SUGAR GROVE STATION, W.Va.—If you’re looking to start a religious order, set up a high-security summer camp—or truly drop off the grid—a secluded Appalachian getaway being auctioned off on behalf of the U.S. military might be just the thing.

Sugar Grove, an 80-home former naval base nestled on 123 acres in the rolling hills of rural West Virginia, is a self-contained community with tennis courts, a baseball diamond, and its own water and power systems. It also offers status-conscious buyers something unique to brag about—a...