If we were to build an actual multi-generational fallout 'vault', what would be the main challenges?

In your example, upper Carlsbad actually isn't dry, it's just pervious to water. Carlsbad is essentially a giant chunk of limestone; the water seeps through the surface soil, the limestone, the upper elements of the cave, and into the lower elements. Water absorbs minerals from the soil and limestone along the way, evaporates in the lower elements of the cave, and deposits the minerals it carried (in this case highly irradiated minerals).

If you build inside Carlsbad, you disrupt this process. Assuming you can adaquately ventilate the lower portion of the cave to facilitate the evaporation of water, you would still have to worry about exposing yourself to minerals as they travel through the upper elements of the cave.

There's also an issue with using some sort of greenhouse cover for the crops because materials that transmit light tend to have a density that is low enough to transmit radiation as well.

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