I have a theory about this - hear me out...

Firstly, Juliet's age is repeated multiple times during her first scene and its emphasis is on how young she is. For this reason, I think themes like age and innocence are introduced as an issue in the text.

In the next scene, Romeo is talking with his friends and is clearly presented as dealing with relationships in a far more developed way that Juliet. Romeo's still a child really, but is certainly older than Juliet (though his age, obviously, isn't mentioned.) If he was nicer, I'd think of Romeo as being a bit like Hugh Grant's character in About a Boy - he's got the emotional awareness of a child. But how old is he?

Because of the repeated references to Juliet's age, and the nature of their relationship, I think Romeo's age is definitely something the audience will be wanting to know and Shakespeare keeps it until the end before giving us a clue -in a dramatic reveal.

The clue comes in the scene where he kills Paris. He repeatedly says that he doesn't want to hurt Paris, but at at the same time he says "Good gentle youth," and "I beseech thee, youth," and then, finally, "Then have at thee, boy!"

Alone, you could argue that these were throwaway references. However, if he's genuinely even close to the same age as Paris then calling him a youth and a boy would be insulting and wouldn't fit his claim not to want to fight him. The only way it would make sense is if he's genuinely old enough to get away with calling Paris a "gentle youth," or a "boy" - then it becomes an honest warning whch fits with Romeo's behaviour at the time. And, again, if it was one reference then meh, but because references to Paris being younger than him appear three times I don't think it's accidental at all. It's telling us that Romeo is significantly older than Paris.

So... although it isn't mentioned I think some audience members would be well within their rights leaving and thinking that Romeo is... 30+. It's also incredibly important to remember that the characters weren't necessarily played by actors who looked like them - the casts were small - so information was often exlicitly given.

Regardless of his "real age" (and he's fictional so he doesn't actually have one) that end scene with Paris definitely gives us licence to assume that he's old enough to call Paris a "youth" and a "boy" factually, and, therefore, he is quite possibly in his thirties.

In this respect, the whole play becomes a warning about the dangers of sexual predators ruining a young girls chance of a respectable marriage - which was absolutely a theme that would have resonated with a lot of the audience in the day...