You could build a raspberry pi and add RetroPie and emulation station. The raspberry pi and the rest of what you need should go for less than $100. If you want more space or ram, I believe you can pay more for a better board.
But it's the cheapest way to build and emulation box or device.
A raspberry pi is essentially a compact PC, that you can buy. You can use it like a real PC if you want as well. Usually it's sold as just a board. You will need to buy or build your own case, and buy a power chord, mouse, keyboard etc. It's really fun to build but is great to use as a PC.
I had two SD cards, one with an OS, to surf the web and such. And I also had another SD card with the emulators on it. I just switched them out whenever I wanted to play or surf the web.