Buy yourself a new gaming rig, and use your old gaming rig as a server. That's what I usually do.
Or, see if you can get an old office PC for a couple hundred bucks on eBay. Anything that's around 5 years old (10 is pushing it) and has decent specs (maybe an i7 and 16GB of RAM) should work fine as a Minecraft and Plex server. Then you can get a cheap (ideally less than $200) graphics card and be good to go.
Bottom line, a "server" is just a PC that's serving things. You don't need enterprise grade hardware. If you're new to hosting, I'd advise you to start cheap and then upgrade to better hardware in a few years when you KNOW what you need. No need to get something really nice and expensive right now.