I guess in "theory" you could probably find a 2000 watt PSU, and a custom case that will fit multiple computers, but you really are talking expensive right now. To give you A hint, if you look at my signature, the first computer listed, has eyfinity, and can handle 3 seperate monitors at once. (keyword is 3, not 6) If I was to setup something along the lines of what your after (to do it right) here is a basic build I would go with.
2 of these video cards ($200):
A crossfire enabled motherboard such as this for $64:
A basic quad core (a dual core will work, but theyre hard to find now) like this for $80:
lets throw in some memory $30
A crappy but usable computer case for $23
A high end power supply (needed if you want it to last) for $90
Throw in a 400 gig hard drive for $20
couple cheap fans and whatever you can find for a optical drive.
throw in a OS and your looking at aroung $6-650 give or take. Keep in mind this is a pretty barebones system, but should in theory give you 6 monitor support. If you can make it work with linux, you can drop 100 bucks off of that, but that would be questionable.
Alternatively, if you used a case similar to that, and a high enough powered PSU and CPU, with a full ATX board you can just throw in 2 of the cheapest video cards you can get and not crossfire them, it will give you 4 monitors, but doing it that way is often a hassle to set up, and is buggy as can be. Depending on how old of a video card you got, you can maybe get away with a 500 watt PSU. Doing it this way though, assuming you could find cheaper parts, I dont see you getting the cost below $300. But either way your going to need a full atx motherboard, a case that will fit it, and a high quality full ATX PSU with a minimum of 500 watts-higher depending on the video cards. To use the setup I suggested your going to want a high end 750 minimum-850 recomended (I have a 850 in the list I suggested)
In any case, id hold off a bit-perhaps someone else here will have a idea of how to do this cheaper, using different hardware, but this is a general idea of what you need. If I were to build a system for what your after, the above is what I would use.
Primary system: Motherboard: ASUS M4A89GTD PRO/USB3, Processor: AMD Phenom II x4 945, Memory: 16 gigs of Patriot G2 DDR3 1600, Video: AMD Sapphire Nitro R9 380, Storage: 1 WD 500 gig HD, 1 Hitachi 500 gig HD, and Power supply: Coolermaster 750 watt, OS: Windows 10 64 bit.
Media Center: Motherboard: Gigabyte mp61p-S3, Processor: AMD Athlon 64 x2 6000+, Memory: 6 gigs Patriot DDR2 800, Video: Gigabyte GeForce GT730, Storage: 500 gig Hitachi, PSU: Seasonic M1211 620W full modular, OS: Windows 10.
If I don't reply within 24 hours of your reply, feel free to send me a pm.