The old Antec SLK3000-B was a relatively compact (~ 18" tall by <17" deep) ATX case (also holds mATX mobo) with five HDD bays. No longer offered by Antec, but Rosewill has the same case in their R6A series, and HEC/Compucase also sells them under their own name now (6A series). Compucase is the OEM for all of them. The design is excellent with 120mm f/r cooling (almost never available in a mATX case), but I don't see any R6A now on the Egg (Rosewill is an in-house packaging subsidiary of the Egg), so IDK where you'd find any except maybe on fleabay. The Egg has several variants of the 6A under the HEC/Compucase brand - a couple at under $50. +sh.
mATX cases seldom have more than two HDD bays, but you can check the Silverstone Sugo series if you've got more money to spend. Well, you can skip the Sugo as they have only 2 Hdd bays. And you also don't find regular ATX PSU support (which you may want/need to power a stack of HDDs as well as the system at a reasonable price for quality units) in mATX cases.
Using the Newegg Power Search, you can find several mATX cases with 4 HDD bays, but they won't be as flexible or give as good cooling performance as the 6A models. The NZXT Rogue is an interesting one that offers 4 HDD bays, but it's pricey and cooling may be an issue.
Good luck.
.bh.