The two primary characters are the older waiter and the younger waiter. Though the character of the old deaf man is not a central one because of active participation in the dialogue or plot action, he is central because he forms the focal point of the whole story and is of pivotal importance even after he goes home. We can set aside the problems of the soldier, girl and bodega barman and focus on these three central characters.
The old man's problem is clearly defined by the older waiter: He is old, deaf, alone in the world since his wife died and, though he has "plenty of money," he has "despair." This despair is not a despair related solely to loneliness and old age but rather is related to the existential condition of the world that his wife, the older waiter suggests, helped him stave off and keep subdued.
The existential condition of the world is that it has no meaning and no order. Each individual, in order to live with some peace in a world tending to entropy, death and destruction, must find or create their own meaning and orderly mode of existence. Tthe old man's order and meaning were bound up, as suggested, in his wife; now that she is gone, even "plenty of money" contributes nothing toward meaningfulness and order (except to buy his seat and brandy at the clean and well-lighted cafe).
The young man's problem is that his job keeps him up and away from home until the wee hours of the morning; he never gets home until 3:00 a.m. He has a wife whom he never sees yet who waits for him nightly in their bed. This is his problem: youth and desire interrupted by the necessity of work.
The older waiter's problem is that he feels the existential meaninglessness and disorder of life and has only his job to keep away nihilistic despair over this lack of order and meaning in life. His feelings of existential suffering keep him involved in providing order and meaning to others by providing a clean, well-lighted place for them to rest for a while. His existential feelings also keep him awake at night with what he naively calls "insomnia." He is on the brink of nihilistic despair of the same sort the old man experienced "last week."
Now, without thinking further, he would go home to his room. He would lie in the bed and finally, with daylight, he would go to sleep. After all, he said to himself, it's probably only insomnia. Many must have it.