Star Trek roleplaying
Hi! I've done a lot of thinking on how to handle a Star Trek game, but my thoughts usually run more towards Next Generation era games. For those, I'd recommend a very gentle hack of the Leverage RPG from MWP. Just change the Roles to Departments and you're good to go.
While that might work for an Original Series era game, you may be better off using the Firefly rules. Unfortunately, that game might take a bit more work to hack, but not much. Just tweak the skills list a little and you should be more or less good to go.
The problem with those is that they aren't as free as Far Trek, Where No Man Has Gone Before, and Lasers & Feelings. Still, they're designed to run TV-style adventures in those styles pretty effectively.