No. You will not get .001 accuracy for $500. You can blow $500 on motors, drives, and power supply. Tight tolerances need accurate slides, and lead screws. Ball screws can go $200 with bearings. Couplers go for $30-$60 (for the project). Three good zero backlash lead nuts for acme lead screws can run $60. A good lead screw can run $45.
The sad truth is, this is one of those hobbies where if you buy cheap, you get the frustrations of buying cheap. You lose so much time trying to work with low grade/under power systems, you might give up before you got off the ground.
If you are stellar at scrounging, if you find the best deals, if you accept middling performance, you might get away with $800. the basic problem is that a 12 x 12 uses all the expensive bits a 24 x 24 or even 24 x 48 uses.
I've heard some claim it's cheaper and easier to just pay for then fast prototypers,