As he headed for a private bunker beneath a mountain in suburban Seoul, Woo Seung-yep ran through his mental checklist: canned food, chlorine bleach, radio, gas mask.

Since quitting his job as an information technology manager five years ago, Mr. Woo has been preparing for war.

Most of South Korea’s 51 million people tend to shrug off the North’s missile launches and nuclear tests—they have lived with the threat for years on the divided peninsula. Some 58% believe there is no possibility that North Korea would cause a war, according to a Gallup Korea survey released this month, up from 53% in 2014.

Others aren’t taking any chances.