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The Dead Sea, rivers in California and the Aral Sea are said to be shrinking due to water usage, e.g. for agriculture. Yet the water must go somewhere.

After the fields are irrigated, the water either soaks into the ground or evaporates. In the first case, the groundwater is replenished. In the second case, the water ends up as rain, presumably within a few hundred kilometers of the evaporation point. Either way, the water gets back in circulation.

So what is happening here? How can water be "used up"?

You're making a mistake, at least for the second case:

In the second case, the water ends up as rain, presumably within a few hundred kilometers of the evaporation point.

You cannot model a dry region (or indeed any region on earth) as a closed system for hydrological purposes. When water evaporates in a dry climate, it transports much farther than a few hundred kilometres. General circulation can transport airmasses for thousands of kilometres. In all likelihood, when the water finally precipitates it will do so in a different catchment area and/or far upstream, often in an area that already has plenty of precipitation. From there it may flow thousands of kms to different climate areas yet. This is why in particular hydro lakes in hot climates have such a large impact on ecology: a hydro lake is far larger evaporation than a river, due to its much larger surface area and other factors. When water from a dry region is gone, it can, for all practical purposes, be counted as a loss.

This is a really complex problem and would require a really detailed explanation about atmospheric circulation, meteorology and hydrology.

The short answer to your question is that water is going somewhere else. If you look at the studies about moisture recycling, among the others van der Ent et al 2014 or 2010, you can see that the precipitation that generates from locally generated moisture is just a minor fraction, and usually is higher in very wet areas. So, your assumption of the "few hundreds kilometers" is generally wrong. Considering a very wet area like the Amazon forest, you can see that the moisture locally generated is the major source (about 70% in quantity) of the precipitations in South-Eastern Brazil. This is due to the atmospheric circulation and the topography of the continent.

Source van der Ent et al 2010.

To fully answer your question, both in case of precipitation and in the case of underground percolation, the water used locally in agriculture or other uses, would go to contribute to other drainage basins (both in the cases of surface water and underground aquifers).

Most of the world's land is more than one hundred kilometers from the nearest ocean, yet still gets rain. There are portions of the Rockies that are about a thousand kilometers from the nearest ocean. And since clearly a large portion of the precipitation that lands in the Rockies flows to the ocean, there must be some mechanism taking water back to the Rockies from the ocean.

Furthermore, "several hundred kilometers" is a rather large distance. The Dead Sea is only about 50 km long and 15 km wide with an area of 605 km^2; if you draw a circle of radius 100 km, the Dead Sea would make up about 2% of that region. 100 km from the Dead Sea gets you to the Mediterranean, and 200 km gets you to Egypt, Saudi Arabia, Syria, or Lebanon. So even if water were to stay within a few hundred kilometers, it would be unlikely to make its way back to the Dead Sea.