Starfield does not include a multiplayer mode, either competitive or co-op. That's the norm for Bethesda RPGs: The multiplayer Fallout 76 is an anomaly, while the rest are singleplayer-only adventures.
Flying around space with friends sure does sound like fun, though. It sounds like so much fun that people have pledged $600 million toward the development of multiplayer space sim Star Citizen, and that makes me think there's a decent possibility that Starfield multiplayer is in our future—just unofficially.
It took a long time, but modders managed to get multiplayer working in Skyrim with the Skyrim Together Reborn project. Lauren tried Skyrim multiplayer when the mod released, and it really does work. It's janky, yeah, but so are Bethesda RPGs when you're not using a mod to add co-op, so that's nothing new. (At the moment, people's Starfield ship cockpits are flying away without the rest of the ship.)
The best Starfield mods right now are pretty basic, but as time goes on and new official and community tools are released, the complexity will ramp up. Bethesda games have some of the biggest mod scenes out there—Skyrim modders have used the RPG as a platform on which to make whole new games—and I expect we'll see the same creative energy from Starfield modders.
In the meantime, if you're itching for some multiplayer space adventuring, No Man's Sky is still going strong.