

The obvious answers are the games we endlessly replayed historically: Mario Kart, Goldeneye (VS mode), Halo (VS), Smash Bros.
If you specifically want ones on PC, I’d suggest Starcraft, Age of Empires, and probably Counter Strike (I wasn’t into that one, but it had a huge following).
Many board games fit the bill as well. Codenames (physical or online at horsepaste.com) comes to mind, and another commenter also mentioned chess.
Basically any games that were made before endlessly grinding became a thing (yep, that’s only been a thing for a decade or two).
I’m using my old desktop from 2010. There’s no such thing as a server that can “do it all”, but any computer from the last 10 years would probably be a fine place to start. The more you do, the more likely you’ll be to hit some sort of performance limit, and by that time you’ll know more about what you actually want.
In short, find old cheap/free hardware and start playing around.