Frankly, I think this would be very difficult to pull off.
You could give all the characters made-up names that don't come from any nationality. But this has two catches: (a) One or two characters with unusual, made-up names would be plausible. Like if the characters are named Bob and Sally and Mamber and Fred, a reader might think, hmm, Mamber, what an odd name, but there are people all over with odd names, it wouldn't be implausible. But if all the characters have odd, made-up names, the reader will surely find this unlikely and disconcerting. Then the story isn't set "anywhere", but in a make-believe place. (b) In any case, made-up names are still likely to have the "sound" of some nationality. Like if I say that a character is named "Komomuro", I just made that name up, but I think most Americans, at least, would say that it "sounds Japanese". Every now and then I hear of corporations spending millions of dollars trying to come up with a name for a product that will sound "normal" in many different countries. It's not easy.
You could have all the characters go by nicknames -- "Baldy" and "Red" and so on. But that tends to sound odd, too. In real life most people DON'T go by descriptive nicknames. And the nicknames are likely to be place-specific. Do people in Zimbabwe call each other "Red"? I don't know.
You could avoid giving the characters names at all. Just call them "the tall man" and "the red-haired girl" and so on. Possible, but I think that would get tedious and awkward pretty quick.
You could have names from a mix of nationalities. Have one character named "Jones" and another named "Wang" and another named "Lopez" and so on. Possible. It would certainly make it unclear where the story is set. If there are only 3 or 4 characters that could work, but beyond that it starts to sound weird also.
Frankly I think the best solution is more along the lines of Filip's: you can discuss universal themes while still setting the story in a specific place. People all over the world can relate to, say, Romeo and Juliet. I'll bet many don't even remember where it's set. ("In fair Verona", right?, but whatever.)
It would be extremely difficult to write a story that gives absolutely no clues to where it's set. You'd have to carefully purge every conceivable cultural reference. If you say, "Bob wore a blue shirt", then this must be in a culture where men wear shirts and not, say, togas. "She drove home." A society where people have cars, and not just any people, but people of the social and economic class that this character is a member of. And women are allowed to drive, so apparently not Saudi Arabia, etc. Trying to write a story where everything that every character does is a plausible thing for someone to do in any society anywhere in the world would be daunting, even if you know the cultural norms of every society in the world. Odds are you don't -- I certainly don't. I'm sure that if I tried to write a story set in some place I've never been to, it would be filled with errors that people there would find jarring or at least amusing. Ranging from big to small, from, "Umm, your whole story is about these two people dating, but people here don't go on dates. Marriages are arranged by parents" to "You say the hero lit a cigarette during the meeting, but smoking is not allowed in any public place in America any more".
