Mini ITX is 6.7"x6.7", 4X 6.7" = 26.8", a 19" rack is... well you see the problem now I hope
You need to think outside the box. Literally
Leave the top of the rack and mount the boards vertically!
Assuming you can't stack the boards on top of one another, (and you could only do that in a 1U enclosure if there were no components on the boards methinks) you could theoretically fit in a 2x2 configuration - assuming 1 centimetre* gap between each board that'd give you a square of 17+17+1 = 35cm. This is theoretically fittable into a ~44x60cm pizza box but doesn't leave much in the way of space for storage, cables or power supplies. 80cm deep racks give you a bit more breathing room but not a great deal; don't think I've seen any signs of anyone going so far as to try making one, especially in a 1U format. Might be doable with boards that take DC in and only use USB/mSATA/M2 storage and maybe an internal KVM (else you'd also need to route all the USB/PS2/VGA etc for each board out to the back) but I don't think it's a goer on anything else.
Makes more sense to go for a platform specifically designed for high density, like blades or that weird hybrid format intel came out with. Whole lot of space would be wasted using mITX since it devotes a fair amount of real estate to the PCIe slot in any case.
* UKian here and I want to get as much usage out of my beloved SI units before the bonkers nationalists come insist on us re-adopting the imperial system, or something even more ludicrous like beard-seconds.
D'oh, ninja'd by Patrick. Multitasking is a lie perpetuated by those who won't admit to doing a whole lot of things slower!