Unfortunately I'm not familiar with the US market, but in Europe we have sites like Geizhals ("Skinflint" in the UK) that are excellent at listing electronics, so you could source them for cheap: internal drives, external drives
If whatever you end up doesn't have enough 3.5" slots, you can get these things on ebay to use them as a hacked together external housing:
You'd need to get the SATA cables out of your case though.
As for what drives to use: If you don't need redundancy/parity, then a single 12 (14, 16, 18, whatever you need) TB SATA drive will probably beat everything else pricewise. I'd say that leaves you with roughly $300 for the system itself, if you need to buy a new one.
but doesn’t have any way of adding a bunch of drives
Well, you only really need one or two drives. Are you sure it doesn't offer any SATA connections?
What about PCIe? You could use a cheap HBA card then.