Are there any ATX HTPC or desktop style cases with at least 3 external bays and can fit a 120mm tower cooler? Specifically the Xigmatek Gaia, since that's what I have.

That seems like a contradiction in terms. You don't want a tower style case? Generally any of those have more than enough room for the cooler you describe. If you are going mini-ATX, you might be able to find one if you do not install a DVD/Blu-ray drive in it. I have a Antec NSK1380, which at 8" tall, might just fit it, again if you do not install a drive. Otherwise, I don't think many of the smaller cases are designed to have that many external drives and/or be able to fit those style coolers. Sorry I cant be of much more help. Do you have a list of components you want to put into this machine? That might help us give you more information.

Only one I can think of with 3 5.25 bays is a CoolerMaster CM Media 280, but that has been discontinued for several years. I looks pretty bad as well to be honest (I own one, but replaced with a Antec Fusion Remote Max a couple years ago). Since it has a 120mm rear fan in the standard location (above the motherboard i/o area), you can pretty much fit any tower coolers out there in that case.

http://www.coolermaster.com/product.php?product_id=150

That said, good luck finding one... Only place I can think of would be ebay or craigslist and see if someone is selling their used one.

Still haven't found anything, though it occurred to me a 4u rackmount case might work. Been searching but haven't found one I really like. Ideally it wouldn't have a front door and wouldn't be too deep. The shelf it will be on is only 17" deep. I already cut most of the back out so airflow won't be a problem even if the back of the case is flush with the back of the shelf but I wouldn't want it to hang out over the front by more than 1" or so.

3 external 5.25"s is not that common except in Rack Mounts.

I built a silent system in a Lian Li PC-C32B 4u Rackmount case
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