The technical term seems to be a JBOD bay. (Just a Bunch Of Disks)
Basic ones are probably usb, ideally you have something that has a SFF port. Modern ones might also have thunderbolt.
Finding a micropc that supports SFF out of the box might be a challenge but some do support pci express cards.
Apparently there also exists something like Oculink which is pci over cable but i know even less about that one.
EDIT: if you look for “Nas enclosure 4bay” you actually do find plenty of options (Jonsbro N3 per example) that allow you to build it all in one unit with a mini-itx board. A nas pretty much just is a pc with special software so this would be what i recommend.