Kind of my thought. A Dell PowerEdge T430 looks like it would fit the bill nicely--up to 8 3.5" drive bays, plenty of cores, plenty of RAM, in what used to be called a mid-tower chassis. Still getting firmware updates, decent (albeit kind of slow) remote management, plenty of PCIe slots for expansion if needed, redundant PSUs, etc. All decent server-grade hardware, even if not the latest generation--but still recent enough to support PCIe bifurcation. It's not going to be super low-power--2x Xeon E5s will do that--but plenty of capacity to do what OP needs, and within budget. US$453 here, including a pair of E5-2620s and 32GB of RAM--no drives, but drive trays for all eight bays:
I'd probably bump up both the CPUs and the RAM, but it's a starting point.
Edit: Really, you could get this server and a
pair of Exos 20 TB disks for under $1200--a little over the stated budget, yes, but that includes the storage. And there's still room for six more disks in the server. Mirrors aren't the most space-efficient way to set up storage, but they do make expansion easier.