If CPU speed is not particularly important, I suggest getting some Wyse 5060 thin clients. You can get these for around $30 per unit on eBay.
Then upgrade the storage and RAM. They can take up to 2x 8GB DDR3 SODIMMs. Storage is a SATA DOM. You can just remove the board from a lot of 2.5" SSDs and use this. I've done this with a 500GB Samsung EVO 850 and it was pretty easy.