If you want to save money, a four player cab ain't the best way. You'll be spending more on the wood and laminate than $50 as raungst stated (closer to $100+.) Then to get everything to work together nicely, you're going to need more than just a Pi.
Ditch the Pi and cup holders, use part of those savings to pay for the wood and a cheap encoder and find someone to give you an old computer that was just going to be tossed anyway.
Or better yet, re-evaluate your priorities. I've seen cabinets done inexpensively, but more often than not, a lot of the materials/hardware was already on-hand. Starting from scratch, it's going to be really difficult to get satisfying results from a really tight budget.