Most likely it won't need anything other than some very basic firmware which would be part of the emulator.
It is unlikely to need any mechanism for managing drivers for example (unless you can build custom hardware).
You might need some form of scheduler if you are running different programs to do different stuff (weapons , engines , navigation) but this could be something very simple and again baked into the emulator.
Some form of shell might be nice, but again not necessarily required.
You could run your spaceship computer like that, but you could also install a multitasking system that allows you to download files from your favorite space BBS while playing a game of solitaire and plotting the course to your next mining asteroid.
It will not run Linux, because Linux requires a 32-bit CPU with megabytes of RAM. Someone might write a Linux-like system for it though.