Somebody actually build a PC inside a Ikea Dröna box insert / drawer. I think I saw it on r/mffpc. Was pondering the same before, but inside the (apparently now outphased) plastic Lekman box. The Dröna is a mesh-wire frame Kallax / Expedit box insert. There are also others, including a half-height box with semi-transparent walls (IDK its name, but I've seen it on the ikea site a few months ago when digging into Kallax options for my room, as a replacement of a too clunky, oversized desk).
Edit: If you got access to a 3d printer, you may be in luck as well, somebody did indeed
create a whole case specifically for the kallax shelf.
Or just build the system right into the shelf,
like this guy did for his wife 
If ATX is the only way to go, budget cases I stumbled over in the last few months:
-
Chieftec UNI UK-02B / W - you might have to remove its feet to make it fit (snuggly), but as it neither got a bottom intake nor top exhaust, that shouldnt be too much of an issue
- Jonsbo RM2 - same issue, you probably have to remove the feet to make it fit snuggly
- thats it (at least in European markets)
Other options include, similar to like others already suggested, modifying an existing case - but here, one would go for a case that originally wasnt meant to be used for ATX. Eg. if you could find yourself a used Meshlicious, you could just do the mods
suggested in my Mesh-mATX build log (based upon the original work by SG Custom), which are basically identical for ATX, and then add a drive cage or mounting options to the other side; although with an ITX GPU, you could still use the top GPU mount and just add the HDDs below.
But really, you'd be better off by switching down the mainboard size at least to mATX.
Then you could also just pick a Silverstone SG11 and be happy, because it already comes with NAS basics (eg. a HDD frame for 3x 3.5 / 2.5" in the front with 120 mm fan cooling option). Or just step up the game with the SilverStone Case Storage CS351, which is basically the SG11 but on NAS steroids (comes with a 5x SAS/SATA 3.5" hot swap drive cage).
The SAMA IM01 / Inter-Tech Pocket IM-I fits easily inside a Kallax shelf, but its not so nice for adding massive amounts of drives; but there are a few threads across the interwebs and reddit that show off creative solutions for that issue (eg. 2x2 HDD drive cages mounted to the side panel plate).
cu, w0lf.
