The MiSTer FPGA project is fun and easy to get started with.
You need:
1. The FPGA board from Terasic (Linux ARM host, has USB, HDMI) for $130
https://www.digikey.com/product-detail/en/terasic-inc/P0496/...
2. To run the cores for the vast majority of supported systems you need a custom 32 MB SDRAM board. For timing reasons the DDR3 memory on the FPGA board can't be used.
https://github.com/MiSTer-devel/Main_MiSTer/wiki/SDRAM-Board
I bought mine for about $30 from someone in the MiSTer forum - there are a bunch of hobbyists that sell them there.
3. Some of the more complex cores (Amiga etc) need a custom IO board (VGA, SD card slot, etc):
https://github.com/MiSTer-devel/Main_MiSTer/wiki/IO-Board
This one also seems to go for around $30 in the forum.