Isn't Star Trek about everyone basically working for the government (ie, in the military)?

No; Star Trek focuses on the "military" branch (Starfleet), but the majority of the population of United Federation of Planets are civilians.

I stand corrected. Picard's home comes to mind.

There has never been an attempt at a television series depicting Star Trek outisde Starfleet, correct? I mean, something created by the makers of Star Trek.

I don't know of any. Deep Space Nine came the closest, especially before the Dominion War arc. While the station itself was commanded by a Starfleet officer, it was still a sovereign Bajoran territory and most of the inhabitants were civilians.

Well, everyone apparently works for free under the Roddenberry vision, so figure that one out. Presumably if corporations aren't controlling things like the Federation database, it would seem likely a significant portion of all real development is going straight to the government though, yeah.

Oh, the Federation, where the world isn't run by corporations? Yeah, Google is not working on that future.

Google is working on turning sci-fi technology into reality, not sci-fi politics. In the absence of a Third World War, a warp flight and subsequent first contact with the Vulcans, leveraging existing economy seems to be the best strategy for that.