Does the main character in a short story need a name? My story gives lots of other details about the character.
Can you do this? Of course. There's no law against it. The Novel Police will not break down your door and arrest you for failing to give your hero a name.
But why do you want to?
If you have some good reason, then sure. Like, My protagonist represents all humanity, and so he has no name because he is a generic person and not an individual. Or, My protagonist has no name because, as I tell the reader in the first paragraph, he is keeping his identity a secret. Etc.
But if your reason is, Because I saw it in this other story and it seemed like a cool idea. Or, Because I can't think of a good name. I'd say, don't. Make up a name.
There are plenty of times when it is a good idea to break the conventional rules. Break the rules when it serves to make a point. Break the rules when following the rules creates a problem. But don't break the rules just so you can say, Look! I'm creative and original! I broke the rules! This almost never works.
Important characters in many books are left unnamed, but (in my opinion) it's a bit risky. Unless your narrative is gripping and fast enough to divert the attention of the reader away from the name, or lack thereof, it feels fake. 'Fight Club, for example, is a splendid example of an unnamed protagonist. You could also call your protagonist by their profession or a trait, if you find you need other characters to call out to your protagonist.
Check this out, it might be a similar question
You can manage without one, but I'd recommend against it, unless you have a compelling reason.
A protagonist without a name is often awkward to write, and certainly awkward to think or speak about. A character's name is a pretty basic, fundamental aspect of their identity; there's something weird and forced about deliberately avoiding revealing his name. There's also the simple fact that the reader expects to get a name; if the reader doesn't get one, he's going to think that's a little odd.
These are all things you can overcome, certainly in a short story that can stand a little weirdness. That might be worth doing - if you have a good reason for it.
That's what's happening in the examples some of the others have given:
- In Rebecca, the protagonist has no name; she's only known as "the second Mrs. De Winter." That emphasizes how her own name has become unimportant; her identity is being subsumed in her marriage, how she's constantly being compared and contrasted to Rebecca.
- In Fight Club, there's good reason to conceal the protagonist's name - there's a major twist involving it at the climax. It also represents how the character feels hollow, fading away, a nobody.
Those are good reasons to leave a character nameless. If you have a good reason, a short story can certainly bear that weight. But if you just don't have a name for them, or if having a nameless character is just an arbitrary decision, then you're probably better off choosing some name and avoiding the issue.
It's a lot easier to think of the character if s/he has a name, but it isn't strictly required. In Daphne DuMaurier's Rebecca, the first-person narrator is never named, and is only known as "the second Mrs. De Winter."
Some of the most beloved stories are fairy tales where the characters are not named. They're the princess, the wicked witch, the boy, the big bad wolf, and so on. (Aimee Bender writes some fantastic stories with unnamed characters.)
So no, names are not necessary. However, you'll have to evaluate for yourself whether naming your own characters will work better for your story.