I've been wondering this for a while. In the story I have planned, I've been referring to them as "the boy," "the girl," etc. Just a few days ago, I gave all my characters names. Now it just seems weird. So, what does every one think the benefits of having nameless characters be? I'm contemplating making them nameless.
Well, having used this technique in text-based roleplays, I can say using nameless characters is a double-edged sword. On one hand, it keeps things vague, allows you to experiment, and prevents the reader from really making a connection to the character. On the other, there ends up being a lot of repeated pronouns, the ambiguity can make it hard for the author to figure out what is the character's "personality" and the "mysterious stranger" bit can easily seem cheesy if not done correctly. In the end I ended up naming my character cause I got tired of trying to write "he" and "him" in creative syntax, but if you think you can do it well and if a nameless character fits your story, then I'd say go for it.
I'll answer the question with two examples. Joyce's short story "Counterparts" begins with a character who's named Farrington. At the end of the story, however, his name is replaced by "the man." In this particular story, it shows that Farrington's problems aren't unique to him, and that any man in the city could be presented with the same problems. So, there's a universality that Joyce is going for. Also, in Wells' The Time Machine, the main character is referred to as "the time traveler." His name is never mentioned. In this case, it has to do with uniqueness rather than universality. He's the only time traveler in the world, and the unique name given to him shows that.
If they're your only two characters, why not? A name only serves to distinguish one from another, and if there are only two, and one is a boy and the other a girl, they don't really need names if you don't like them with names. The trouble I have with names is that sometimes I can't find a name that fits. Giving them a name would detract from what else I've built up with that character. Write it without names, and if it works, run with it.
To me, a nameless character is someone who is either void of a personality or personal life of any kind (think The Operative from Serenity or the Agents from Men in Black) or are defined more by their title than their name, such as referring to a person as The King or The Chief and never referring to their name. But that's just my opinion and it doesn't mean that it can't be changed in the circumstances presented in a story.
As thirdwind said, it has been done before in respectable literature. Another example is Cormac McCarthy's The Road, in which the major characters are referred to only as "the man" and "the boy". It can work, and it has worked. But I think that it looks like a gimmick. I'd advise against doing it unless you're VERY confident, and you have a really good reason.
One of my more prominent characters went roughly 10,000 words into the story before being named. She felt a bit more mysterious when she was going around unnamed (the woman, she, her, etc), a bit more apathetic. But once she was named, it seemed to bring with it more human qualities. I guess in the end it depends on what you're going for, how emotionally attached you want your readers to become. I remember from reading The Road, that in the end, I really didn't care about what happened to either of the characters - not sure if this had to do with them being unnamed or not.
Have a good reason for it. Not being able to come up with a good name isn't a good reason. A good reason could be like in Fight Club. The main character is nameless, because he is suffering from a modern era identity crisis. His alter ego, Tyler Durden, has both a first and a last name, illustrating that he has a much stronger identity.
If it's first person you can get away with it for a lot lot longer. If other people don't know your character's name, and the 1st person narrator isn't going to think it any time soon, you can play a nameless character for a long way through the story without really bothering the reader too much. On the other hand, I read a story that had a narrator who didn't know their own name, and people kept giving her names - like, 5 of them for different sets of characters who met her, and that was really annoying. It made her even more... spread out and nameless? It was harder to identify with her when one page she was this name, the next she was that one. It basically distanced her from the concept of having a name even more than actually not having one in the narration. Just something to think about.
HorusEye beat me to it, but it's a point worth repeating. Have a good reason behind your choice. Names are easier both for writer and reader to identify with characters, so if you choose to leave your characters anonymous, do it for a solid reason. H. G. Wells main character in The Time Machine is anonymous because he is meant to be an objective observer of the future. Personality is left out of it as much as possible, although his distaste and outrage for our eventual destiny is too profound to conceal. The novel is more social commentary than adventure story, and the anonymity of the time traveller emphasizes it. Any time you are tempted to break convention, make sure you have a better reason than "because it is different, and will make my story stand out." Carving "Jesus Saves" in your forehead with a scalpel will make you stand out, but people will look upon you as a freak instead of giving your message serious thought.
The first person narrator in Daphne Du Maurier's Rebecca is never named--well, she's only 'the second Mrs De Winter'. It fits in with her feeling like an out-of-place nonentity, and keeps the evil Rebecca the main attraction. Like Cogito says, there should really be a reason for not naming the character.
