Austin Harris thought his years as an investment banker had taught him all about tough job interviews—until one that started with a daybreak run though New York City’s Central Park and ended in a cramped office gym for a round of pull-ups, push-ups, squats and burpees.

It wasn’t easy keeping up with his prospective employer, a former college football player and co-founder of a company that makes nutrition bars, said Mr. Harris, 35 years old. When they started talking business, he said, “I was breathing pretty heavily and trying...