A Star Trek RPG similar to Mass Effect could easily be made. There's enough races that could be used as antagonists threatening the known galaxy -- the Borg, Tholian Assembly, Breen but most interesting would be Species 8472. The player character could be some 'veteran' or 'newbie' Starfleet officer sent off in a ship travelling around several planets looking for allies & clues on how to fight the 'evil'.
I can see this... kind of, but you'd have to have a one heck of a work around for the main character not having money.
1) You could make a Federation citizen who isn't part of Starfleet.
2) There is currancy in Star Trek and the replicator can't make everything.
3) No, the state does not just hand out ship and weapon parts to anyone who wants one.
4) 'The Federation' is actually a collection of different governments who often disagree with one another. As of the end of DS9, it's also in a state of economic, diplomatic, and military turmoil.
The question isn't if someone could make a good Star Trek RPG. The first time I had any interest in Star War was Kotor I and II because a good developer can take any popular setting and create a good RPG out of it.