In this helpful guide I'm going to go through what I did to make a self-contained Minecraft server at home on the cheap and also keep it low power but can still run 2 or 3 separate worlds at once (depending on the amount of players)

The main thing that bought all of this together is MineOS Turnkey. This linux distribution of Turnkey is ready to deploy with a webGUI for both Minecraft and the OS itself so you won't have to struggle with staring at a terminal all of the time (but you will have to use it sometimes). You'll more than likely be going with the 64-bit version depending on your hardware and speaking of hardware.

Parts!

ASUS Mini ITX AM1I-A

AMD Athlon 5350 APU

Kingston HyperX FURY 8GB Kit (2x4GB) 1600MHz DDR3 CL10 DIMM - Black (HX316C10FBK2/8)

Sentey Mini ITX Ss5-2514 Computer Case

All of this so far brought me up to $222 but I didn't mention a disk drive, you can use really anything but I used a Crucial 128GB M4 SSD for the low noise and low power, though honestly you don't even need 128GB so if you can pick up a cheaper 64GB. With the 128GB SSD in mind everything came together for about $300 total. A HDD with work just fine if you're looking to go as cheap as possible.

On a final note you will need a USB thumb drive to create the boot stick. And an external monitor & keyboard for the very first install. Also I'd recommend getting a Uninterruptible power supply if you plan on leaving this on for other players for long periods of time. As this is so low power even the cheapest one will give you a safety net to power outages and brown outs.

Hardware setup

As this build is straight forward there isn't really anything to point out other than to put it all together and if you've ordered different parts you'll have to go on your own anyway, but here are some photos.