If all you want is a minecraft server, just use http://www.freeminecrafthosting.org , I'm not too sure how it works though.
It was going to be heavily modded on a custom pack. I don't know if that site supports mods.
EDIT: Screw that bullstuff, it requires you to fill out a
free virus download survey.
So, I guess we're back at building a computer. Is it possible to get a really cheap, stuffty, easily upgradable one for under $100? Like a Raspberry Pi, but you can actually exchange parts.