Which is grammatically correct?
I am a girl in my early twenties.
I am a girl in her early twenties.
Grammatically, both versions work; they would be parsed as
That is, you are a single thing: a girl of a particular age range.
That is, you are two things: a girl, and in a particular age range.
Semantically, the two things are equivalent; say whichever feels more natural to you.