For story-telling purposes a problem should be one that must be solved. This usually involves a time element, or what some writers and editors call "a ticking clock." In Jack London's famous story "To Build a Fire" the protagonist is in a situation which causes him a life-or-death problem. He must solve it quickly, within a matter of hours at most, or he will freeze to death. In the popular story "Contents of the Dead Man's Pockets," the man who has gone out on the ledge to retrieve his important paper knows he can't stay out there for more than a couple of hours or he will succumb to vertigo and fall eleven stories to his death.
A situation that does not have to be resolved does not usually make a good story. A man hates his job, but he needs the money and he can't think of what else he might want to do. A man is tired of his wife, but he might put up with her for years because of sheer inertia, or because of the kids, or because of some economic factor.
To make a good story, the writer should think about introducing "a ticking clock." In Kafka's "The Metamorphosis" the protagonist Gregor Samsa hates his job but doesn't do anything about it until he wakes up one morning and finds he has turned into a gigantic cockroach. Now he has a problem that must be solved! In Death of a Salesman, Willy Loman is getting so worn out with his job as a traveling salesman that he can hardly make his weekly trips through New England. But he doesn't try to solve the problem until he finds that he is unable to force himself to drive past Yonkers, just north of Manhattan.
Willy, after a pause: I suddenly couldn't drive any more. The car kept going off onto the shoulder, y'know?
In Tennessee Williams' great play A Streetcar Named Desire, the situation is that Blanche is living with her sister and brother-in-law. There is no serious problem as long as they will put up with her, but Stella is expecting a baby, and then there will be no room for Blanche. Meanwhile Stanley is pressuring her to move out. But she has no money and nowhere to go. It might be said that the unborn baby is the ticking clock. Babies insist on being born when the time comes--and they can often cause acute problems.
Any situation could be defined as a problem for someone. A problem is just a situation that someone does not like and that they think they need to overcome. That means that different situations may or may not be problems for different people.
Let us look at this first on the level of countries as opposed to individual people. In the past few years, Iran has been enriching uranium fairly quickly. This is something that the United States defines as a problem. We do not want them to be able to get a nuclear weapon and so we think this situation needs to be overcome. By contrast, Russia does not think this situation really is a problem and so they are not really trying to overcome it.
On the individual level, we can also see that different people define problems differently. Let us say that you like to rock your chair back on its back legs during class. You do this all the time. You do not think it is a problem because you actually like to do it. Your teacher, however, thinks it is a problem and scolds you often to try to get you to stop. For your teacher, this situation is a problem.
So, to find a situation that could be thought of as a problem, just think of any situation that someone does not like and which that person wants to change, get rid of, or overcome.