It might be beyond the scope of a normal hobbyist project, but writing Verilog or VHDL to drive an FPGA might bring interested people half of the way there. You mentioned Altera design software in another post; I don't know if Xilinx's is any better (I'm guessing not really), but Digilent sells FPGA boards intended for the educational market at a pretty reasonable cost. Powerful enough, I'd reckon, to allow for something of this scope to be built.
I actually considered something like this; there are good dev boards for $150 that can drive a VGA monitor and use USB peripherals. The manufacturer provides stock blocks for you to integrate, so you can inspect them but you don't have to build them from scratch (which is super hard).
Doing it on an FPGA, without extensive handholding and in real-world languages, would be about a year of work in my estimate. This is assuming you build your own CPU (in procedural VHDL, not at a gate level) to implement an existing instruction set, and use the manufacturer's provided memory blocks, video blocks, etc. For reference, an experienced FPGA programmer would take about 2-4 months full-time to emulate something like an NES.
It would be a really good experience, and it's the kind of thing a comp. eng. degree prepares you for (we do a capstone project at my school which is like this). As a bonus, you'll also cover analog electronics (which are infuriating) and as much comp sci. and math as you're willing to take on.
Note that this is exactly how they made the Apollo Guidance Computer, except that it was 2,800 chips, each with two NOR gates. And a giant honking breadboard.
And a giant honking breadboard.