It is an important point that Penelope promised she wouldn't remarry until Telemachus grew a beard, because that would mean he's a man and he could rule. I guess the idea behind was that she would have a man to protect her, but this man wouldn't be King. Unless he's a late bloomer, he couldn't be older than 13 or 14. I dunno how old you should be to rule on your own, but the beard was seen as becoming old enough to be considered a man.
Putting chronology aside, the story starts with Telemachus traveling to find his father, and I guess he was likely 14 or so, still young enough to just starting to grow a beard. The rest of the story is likely the four or five years that followed. At the very end of the story, he can't be older than 20.