The first of the Sunday night stalwarts arrived at Veselka, a 24-hour diner on Second Avenue, after midnight. A man with a cane made his way to a table, supported on the arm of the manager, and ordered a bowl of borscht. “I’ve been coming here almost since it opened,” in the 1950s, Marty de la Rosa said.

On Saturday nights, Veselka is often the last stop for college students and the party crowd; a place to gorge on pierogies and rehash the evening. The crowded dining room resembles a school cafeteria with waiters; the line to the bathroom, a slightly disheveled fashion show.

But for roughly the last decade, Mr. de la Rosa, 82, and three friends have met at the restaurant on Sunday nights. “It’s like family,” he said. “We’re all retired teachers. We have a lot in common.” The four men met at the restaurant as lone diners and started talking. And, often, listening: The ringleader, a man with a thick mustache who did not want to give his name, could hold forth on topics such as the history of Israel for hours.

Soon one of Mr. de la Rosa’s friends arrived, wearing the unofficial uniform of the group, a windbreaker and a baseball cap. “This is a terrible tradition,” said the man, who also asked to stay anonymous. “We stay here until 4 in the morning and the next day is shot, like you’ve been out drinking.”