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Could an entire neighborhood or even an entire city be that crowded?

Sure.

Hong Kong’s Kowloon Walled City was probably the densest neighbourhood in history, with more than 1,000,000 people per sq km. Photograph: Alamy

Kowloon Walled City was 119 times as dense as New York City.Greg Girard

While this example was just 33k people in 6.4 acres, it lasted for 40 years.

The streets and alleyways of the Walled City were narrow.
Most were barely wider than six feet and some were so narrow that one
had to walk sideways through them. A massive network of passageways in
the upper levels also made it possible to travel the distance of the
city without walking on a ground level street.

The Walled City had its own micro-climate,
due to the massive amounts of tubing, wires, and
open gutters snaking through the building. The lower levels were
constantly hot, humid, and damp...Because of the smelly, humid
conditions down below, the rooftops of Kowloon would turn into a
communal hangout during the afternoons and evenings. People would hang
out, do laundry or homework, or practice instruments.

"It was like a
strange, urban garden. There was tons of household refuse. It was a
bit of an eyesore, but compared to the area below, the air was light
and breezy. It was nice to come up there after living and working on
the lower floors."

In other cases, a somewhat less dense set of buildings can cover an entire city.

Delhi, India

Could this be most of the earth?

No.

People need to eat and to do that, you need farmland, oceans/lakes, factories or processing centers, and transportation ways. While you have some wild spaces and waterways accounted for, you still need massive amounts of flat buildable land to grow crops and raise animals for meat. Even if everyone is vegan, you still need an awful lot of space. I've written about hydroponics in small indoor spaces elsewhere.

By 2050 the world’s population will reach 9.1 billion, 34 percent higher than today...It should be possible to meet
the future food and feed demand of the projected world population in
2050 within realistic rates for land and water use expansion and yield
development. However, achieving this will not at all be automatic and
several significant challenges will have to be met. (ref)

Can you feed your world? It depends what the population is. If agencies are projecting that feeding 9 billion people is doable with some work, then you can assume that larger populations will be even harder to feed. Not only are there more people, but there's less land to do it with, because the people take over arable land for housing and for other resources like schools, offices, warehouses, roads, parking, factories, distribution channels, etc.

Air quality: Half the world's oxygen comes from the oceans and the other half "via photosynthesis on land by trees, shrubs, grasses, and other plants." Some sources say the ocean's contribution is even higher, up to 85%. Land plants do contribute, though they don't affect oxygen levels much because of the total volume. Though wild spaces and urban trees are important for carbon dioxide sequestering and to reduce air pollution.

Our atmosphere has such an enormous reserve of oxygen that even if all
fossil fuel reserves, all trees, and all organic matter in soils were
burned, atmospheric oxygen would only drop a few percent.

Tree impacts on important atmospheric trace chemicals such as carbon
dioxide and...air
pollutants (ozone, particulate matter, sulfur dioxide, nitrogen
dioxide, carbon monoxide, and lead) will have greater significant
impacts on human health and environmental quality. Urban forest carbon
sequestration and air pollution removal along with other environmental
impacts of urban forests (e.g., water quality improvement, lower air
temperatures, reduced ultraviolet radiation loads) need to be better
incorporated within local and regional planning efforts to improve
environmental quality and enhance the quality of urban life.(ref)

Power generation: That's a whole other issue and should be a separate question. Wind generation is likely your best bet, as wind will be a factor in keeping your cities from collapsing, so generators around the cities will help deflect it. And you can have small generators on top of the buildings.

Solar won't have much space to work. If the rooftops are the only access to the sky, they're going to be in huge demand for just that, and for gardens. There won't be much space for solar panels. Demand will outstrip ability to generate power. Yes, use what space you can. Some solar is better than no solar. But if these are 10-20 story buildings with businesses, apartments, schools, etc, there just isn't much surface area to work with. Maybe future tech will change this equation some.

Fossil fuels are likely depleted since you have a "far-future" tag and that many people. There are other ways to generate fuel, including from the waste products of that many people, but you need some space to do it.

Assumptions

• Population density according to your description even this might be less dense than your world but I will work with this


• Water to land ration like on earth


• Animals are of no concern

So few things that have to be met (without getting to deep into fiction/fantasy):

• More planets/asteroids available than only the one mentioned


• The top of your building(s)? is covered by wind turbines and solar panels


• Health care is great


• Some of the build upon area is use for food production

The Numbers

Based on Kowloon City Density and the land mass of earth and reserving 20% for food generation. We arrive at a measly 154 quadrillion people inhabiting your world.

Food production in 2009 when accounted for population this was about half a ton per person which would amount to 77 quadrillion metric tons per year or 2 439 435 497 tons per second.

Results/Consequences

this was only food and population but the numbers for water/fluid consumption and power will be equally high. I highly doubt any greenhouse even when sized at one fifth of all land mass an earth would produce these numbers so your civilization should already have a partial dysonsphere or something like that in place to generate the power and food demands of this crowded planet.

It is very unlikely that anyone who has the money would stay there so you'll need a good explanation for that. Considering that you need other planets/asteroids and even partial dysonspheres to get this to work in the first place.

Conclusion

Possible but very unlikely. Except when it is something like a big lower class world where the majority of people have to live because of their limited power/influence/money supply.

First some perspective: Earth's landmass is ~197 million mi². Assuming you are describing the population density of Hong Kong (67,000/mi²), that is a world population of ~13.2 trillion or ~1700 times the current world population.

So, the 7 big obstacles here are water, food, air quality, global heat, durable goods, population growth & power.

The first thing to consider is that this world would not have NEARLY enough natural freshwater. Even at current world populations people are beginning to consume fresh water faster than it replenishes. This means that over 99% of the world's drinking water would have to come from desalinated Ocean water. This would consume tons of power and generate tons of heat, but the planet has enough salt water to do this. (We'd likely need to export the excess salt to other worlds since we would not have enough space for it all).

Second thing to consider is that you need food. You need about 20,000ft² of farmland to sustain a human life with modern technology. If you stack troughs of indoor farmland with synthetic lighting, you can probably compress it down to 3-4 layers worth of indoor farmland for each story of human residents, throw in some excessive GMO manipulation, and you might be able to cram it down to 2000ft² of indoor farmland per person meaning the majority of your world city is still actually indoor farms. Being constantly feed by a massive pipe system of desalinated & recycled water. This is where reality starts to question this model since you would need to feed such a massive amount of these farm with water from oceans, delivering enough water to places that are 1000 miles inland would be questionably doable. So you are hitting the realm of unrealistic, here, but if civilization had enough time to adapt to this density, then assume that the whole world would need a MASSIVE underground pipe system. Even then, in-land areas would need to be very efficient about how they recycle their water once piped in in order for any pseudo realistic pipe system to be able to import the needed volumes of water just to deal with irrigation evaporation.

Third, even with all this indoor plant mass, current vegetation may not be efficient enough at those ratios to produce enough oxygen for everyone, plus all the emissions from industry and daily life would be fatal without a stopgap. Going back to genetic engineering our food, those plants would also need to be breed with air filtration in mind. Basically, you would have to co-mingle the farms and populations as much as possible so that your city grid basically encapsulates residences and businesses in the farmland so that circulating CO2 and O2 could happen as efficiently as possible. By being close to high human density, the plants could be engineered to have massive matablisms so they grow fast, filter lots of CO2, and feed more people per cubic foot. The garbage and sewage system of the city would have to recycle all organics back into the farmland; so, the place would probably stink worse than a medeville city, but could be livable if the crops are properly engineered to assure this symbioses.

Forth, people make heat, desalination makes heat, farm lamps make heat, heat, heat, heat... basically, your society has become a massive self cooking oven; so, you'd need to combat that by not just restoring the ozone layer, but by going in the opposite extreme. You'd have to terraform the upper atmosphere to block so much sunlight that day would become an eternal twylite. At this point our machines do as much to keep us warm as the sun.

Fifth, there is nowhere left to mine resources; so, even what is left underground can not be safely gotten without creating dangerous sinkholes that would destroy the city above. landfills could also no longer exist. If you can't recycle it, you don't make it. On top of that, this planet would need to sustain a massive fleet of freighters constantly pulling in resources from many nearby planets to even begin to approach this level of development, much less sustain it.

Sixth, population growth is no longer an option, at this point human life has become such a burden that laws come into place to control growth including mandatory sterilization, family size limits, and possibly institutionalized genocide. Civil unrest would be unavoidable, meaning governments would need to maintain absolute power over people's lives. Democracy and privacy are both dead, predictive AIs would quell rebellions before they happen, and the common person has no access to weapons or education with which to resist.

Seventh, is power. No current form of power is still an option. All the machines needed to keep people alive mean that each person has a much larger power footprint than we have today. Solar, fossil fuels, etc just don't cut it. For a civilization to reach this point they need to invent an economical solution to cold fusion using the hydrogen available in what's left of the ocean's water. This means that the world would slowly consum it's oceans releasing its oxygen to form that much needed ozone I mentioned earlier and the hydrogen to power the fusion reactors. Eventually, the oceans would begin to disappear, but even at these rates of consumption, this civilization could last a pretty long time.

Lastly, going back to my previous estimation of hong kong population density, this means that the average person accounts for 416 ft². That may not sound like a world superstructure at first, but this means that your 2000 ft² of hi-tech farmland, the place you live, the place you work, the place your kids go to school, your fair share of everything and everywhere you go you needs to all fit into 416 ft². So, I suspect, this would probably look like an average continuous building height of ~8-12 stories. Obviously that would just be an average height; so, you'd have rich, "remote", and ruined areas where the super building is shorter than that, and other areas that are massively built up super structures that are hundreds of stories tall of continuous structure; so, I'd say this is probably the low end of what a world building would look like. But if you were to contiguously build up this whole super building up to the pentacle of what material integrity would allow, you could probably get up to a few quadrillion people at the very high-end assuming you get really good at recycling and importing new materials.

For some reason i read this and i get the feeling it's an dystopian world, I mean how can it not be?

In our petit planet we only cover about 3% of it's area, just saying.

Feasible? Probably, a good idea? Probably not.

"So we will have an Earth size hollow sphere with 3 m (9 feet) thick walls. The volume of all this concrete is 1.54∗10^15cubic meters... that means a square surface mine with 1240 km (770 miles) on a side and 1 km (0.6 miles) deep."

That's a big mine, assuming we'll take the resources from earth. But that is only taking in consideration a hollow sphere, are we covering the oceans too? Are buildings going to have multiple floors (I'd assume yes)?

So to make this more plausible we'd be probably mining planets and asteroids for that to be possible, are animals sheltered in like zoos? Do we even care about animals? Is this a working class planet and titan is the rich people's planet? That would be fun. I'd assume rooftops would be really sought after.

I really like the idea and do really think it's plausible, specially if we have colonized other planets. I'm not sure i got the vibe you wanted but i definitely got a dystopian feeling from what i read.

Hollow sphere earth numbers:
https://www.quora.com/Could-we-build-a-spaceship-larger-than-the-Earth-itself

not physically impossible but why not just burrow into the earth?
It would be easier, also don't forget about food it is hard to grow food without sunlight and if the planet was one big apartment complex that is just so much more food you need seeing as there would be about something trillion people. I say either burrow or build a sky scraper into space several times, it also leaves more space for food.

everywhere you go there are stories and stories of shops, markets, cinemas etc above you. Even at the very top, there is only limited access to the open-sky

Why?

There's actually a very good reason why: That open-sky view is being used for power generation.

It sounds like you want a "high tech" but not "very high tech" situation. Let's suppose these people have near-perfect photovoltaics and can capture 6$kWhm^{-2}$ each day, or 168$kWhm^{-2}$ each month. (Reference) That means that if each person has 80$m^2$ of living space (whether or not that's "crowded" is subjective, I'd call it normal), the roof of the top-story apartment will be generating 161$MWh$ per year.

How's that compare to real-world energy production?
India has 1,300,000,000 people and generates 6,444,000,000$MWh$ per year (see here, and double-check my unit conversions). That's 5$MWh$ per year per capita. For the United States it works out to 72$MWh$ per year per capita.

So far it looks like this could work; your "global building" could be as much as 30 stories tall (deep) provided you're not expecting too many people to have modern amenities like electricity or refrigerators.

But where does the food come from? For that matter, where does the oxygen come from? CO$_2$ concentrations can be a problem even in existing buildings.

I think your best bet for a plausible world would be one with much less land-mass than ours.

You'd basically have a water-world, with people living on barges and civilizations powered by floating wind turbines and eating farmed fish and seaweed, and then there'd be this big island or chain of islands, say the size of Japan or Madagascar, that, as the only firm ground in the entire world, had been developed into a single contiguous 100-story-tall megastructure.
For that matter, does your story even need to be set in the fixed mega-structure? Can you imagine living in a cruise ship the size of a city-state?

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Power is really the only limiting factor. With a virtually unlimited power supply, you can run algae tanks to supply food and process air as well as a waste. You'd need some sort of fusion generator system to produce the required levels. You also need a food processing system that can take said algae and produce faux foods like meat, cheese and bread.

Society would have to be virtually 100% efficient as you don't have the room or materials to waste dumping things like society does now. Everything would be made to last and/or recycle 100%.

You'd need some sort of VR matrix system which would allow people to get out and explore without having to leave home. You can't afford people travelling for holidays. In fact travel needs to be kept as little as possible and only short distances. No flying to Paris for the weekend.

You also need automated system for delivery of goods and the removal of waste. You can't afford people physically going out shopping and bringing goods home with them.

Finally space travel would have to be to and from space elevators perhaps to an orbital ring.