The problem with turning 107, Joe Binder said, is that “you can’t work no more.” Mr. Binder turns 107 on Saturday, but he had the confidence to celebrate early during his regular Wednesday lunch at Mario’s on Arthur Avenue, the main drag in Belmont, a longstanding Italian neighborhood in the Bronx.

Until a few years ago, Mr. Binder was holding down two jobs. By day, he tended the Mario’s parking lot. By night, while working as a waiter of sorts, an occupational hazard upended his routine. “I was working as a house boy in a local gambling joint, serving coffee for the guys playing cards,” he recalled. “But the secondhand cigarette smoke was killing me.”