Confederation Chronicles: A Mobile Base of Operations - part 01

Part of the goal for the setting is to allow a bit of the classic exploration themes pioneered by the original Star Trek series, while still retaining that frontier feel mixed in with the 'fallen empire' premise.

Several shows provide the inspiration for that!

Andromeda

This show is named after the ship called the Andromeda Ascendant. The ship itself is an obscenely powerful remnant from the fallen empire's military might. Dylan Hunt (played by Kevin Sorbo) commands this ridiculously powerful ship of the line that has weapons that can destroy planets, automated defenses, and internal repair and construction facilities called the Andromeda Ascendant. The only reason he has it now, long after the Long Night of the setting, is because he got trapped on the edge of the event horizon of a black hole.

The ship has a female avatar that appears onscreen, as well as in a manufactured humanoid body, affectionately called "Rommie", which allows players to interact with the ship and shout commands to it, while it speaks and expositizes to them.

This high level of technology handwaves the need for a lot of repairs and refitting and resupply (unless it's a plot point for the episode), and rationalizes why such a small crew can pilot the damn thing. It also allows players to skip having to answer to a rigid chain of command and go 'adventuring' in pursuit of their goal.

Blake's 7

Again with the ridiculously powerful ship (the Liberator), with a sentient (though less polite and helpful) computer known as Zen that quite possible caused the deaths of its prior crew, the show has the initial trio of protagonists (Blake, Jenna, and Avon) explore the nature of the ship as much as they spend running away from Federation patrols and running to places they can sabotage or get information.

The alien nature of the ship allows them some tension that Andromeda lacks when on the ship -- except when something happens to Rommie, of course -- because the crew struggles to control it and are forced to interact with Zen when it refuses to do exactly what they say.

Also, the speed of the ship allows them to criss-cross a vast interstellar empire and carry out their little missions of rebellion and revenge.

Farscape

As I was reminded in the comments, there's another type of powerful ship that is also easy on the maintenance and advanced enough to be valuable: Moya the Leviathan from the TV show Farscape. Moya was from a race of living ships that bonded with Pilot from the race of Pilots, and even gave birth to another ship known as Talyn. It has little drones that perform repair and maintenance, and has a personality of its own.

Once again, there is  potential for tension due to the alien nature of the ship and its Pilot, as well as the sense that the ship is alien and isn't entirely understandable in the way that an inanimate ship built by one's own race might be.

Moya had no weapons, by the way, except for the defensive capability to Starburst (jump through a tear in space and time) to other locations. This made it interesting when other ships tracked them and shot at them.

Talyn, another Leviathan and child of Moya, had weapons -- but was more of a genetic experiment on the race of Leviathans that escaped.

Next: examples of larger ships that follow the 'aircraft carrier' role in a campaign.