Pre-Dominiom war when everything is pretty chill. Starfleet kinda seems fun until you realize that you have to basically do military training and all this bullshit to join and then you gotta go die in space half the time. I'd maybe just try and get a runabout or the equivalent and just sorta wander around with some pals and look at totally safe and previously explored space things. I would also open a Volcel Precinct on Risa.
I feel like with FALGSC I could have done well enough in my youth to be one of the random engineers that nod silently while a named character tells them what to do during the intro to a scene, and I'd recalibrate the hell out of some dilithium crystals or run a sub-surface probe like a boss. Probably not the on Enterprise, but on one of the other ships.
Really I'd just be hoping that I'd be included in one of the horny episodes left over from the 60's and get to go to one of the sex planets for shore leave.