I would try 4 dual band routers (i.e. they can use a channel in the 2.4Ghz range and one in the 5Ghz range at the same time), and set them up on different channels to minimize interference, checking for this. Go for the best routers you can get from your budget. Spread them over the room and connect them to an Ethernet switch. Try out different positions to get the most from your hardware.
An example of channels you could choose are channels 1, 6, 11 and 14 for 2.4Ghz as per this section and channels 36, 40, 44 and 48 for 5Ghz (I just chose the lowest non-overlapping channels allowable for the USA as per this). It should not matter how you pair those channels onto the different routers, so for example routers with
- 1 and 36
- 6 and 40
- 11 and 44
- 14 and 48
on 2.4Ghz and 5Ghz respectively.
You all set them up with the same SSID and password, but otherwise nothing special, just as you would for single APs. The clients will choose the best available AP (also switch between them transparently if necessary).
The router/AP you mention in your comment is a single-band 2.4Ghz one. I suggested dual band ones. One hint as to whether what you are looking at is dual or single band is that dual band ones (usually) have two antennas while single band ones (usually) only have one. But check the specifications! Dual band routers should perform as good as one router in one band, and a second in the other, so essentially you could two in one.
I think this could work, but no guarantees. The university WWAN where I study tends to get flaky in full large lecture halls (i.e. 300-400 people) and they probably spent quite some time and money optimizing it.