When Tyler Allen agreed to fork over $3 million in cash for a luxury condominium near Concordia, Kan., he wasn’t attracted by the indoor swimming pool, 17-seat movie theater, or hydroponic vegetable garden.

The real selling point of the 1,820-square-foot apartment: It will be buried 174 feet underground in a decommissioned missile silo sturdy enough to withstand a nuclear attack.

Mr....