You do not need a server to run a game clients server.
I thought about doing this for a game but it becomes very complicated very fast, also as said everything is more expensive for servers even used stuff because they are designed for 24/7/365 run times and this is where the margin is for resellers. Game servers do not require this level of reliability, game server clients are typically not designed to run 24/7/365 without restarting.
Cheap old PC likely would run it fine, good quality newer machine may be able to run the game server and client (when you play) on the same machine.
You do not need more than basic video card for a server, any typical ssd would be more than fast enough, you do need a decent internet up speed.