Danny Miner, a 66-year-old retired chemical plant supervisor, spends most days alone in his Tooele, Utah, apartment, with “Gunsmoke” reruns to keep him company and a phone that rarely rings.

Old age wasn’t supposed to feel this lonely. Mr. Miner married five times, each bride bringing the promise of lifelong companionship. Three unions ended in divorce. Two wives died. Now his legs ache and his balance is faulty, and he’s stopped going to church or meeting friends at the Marine Corps League, a group for former Marines. “I...