Quake will Run on a 386, you need at least a 387 math co-processor, Im sure it needed at least at 8meg of ram.
386DX40
64meg ram
2meg video
40mhz 387 math co-processor
My 386 does 1.8fps in the quake timedemo test, thats fullscreen minus 1, the frame rate does'nt really improve enough to make it playable when you shrink it all the way down to a matchbox size.
He only has 4meg of ram, a 386SX33, no mention of math co-processor, 8 bit graphics card, how the hell do you even run quake in 4bit colour? I cannot imagine Quake playing that smooth on my machine even if the colours were reduced that much, to get the frame rate up Quake simply needs more cpu grunt.
The video has got to be fake.
edit: Found a document that describes video modes.
vid_describemodes
PACK1.DRV - Unknown 1-bit color driver.
PACK4.DRV - Unknown 4-bit color driver.
PACK8.DRV - Unknown 8-bit color driver.
I'll be blown away if Quake improves much, Got to find how to switch modes because these modes might be available but not under the video mode menu, got to go to work I'll give this a try later.