OK, so all the drives were originally internal drives anyway - mostly 3.5" drives, I take it? From your description, I take it that your current PC is full and there's no more bays/motherboard sata ports to use?
And there's a reason you don't copy the data off the old drives onto the new drives?
Not sure about cheap options.
Some approaches come to mind:
The ready made option would be a multi-bay external enclosure like this one:
https://www.overclockers.co.uk/icybox-usb-3.0-10-x-3.5-hard-drive-external-enclosure-hd-13h-bt.html
Not very cheap, but the most straight forward option.
You could replace your current PC's case with a tower-server case and then you would likely need to get one or more add-in SATA cards (unless your current motherboard has lots of ports).
Server cases tend to the expensive side, though. And it seems that large server tower cases are getting rarer.
Using your old PC as a NAS is probably not the way to go if it only has a few SATA ports on the motherboard as you'd need to replace that. If the case for it has a lot of drive bays, you could try
converting it into an enclosure (if it has 5.25" bays, you could also install a hard drive back plane e.g.
https://www.overclockers.co.uk/icybox-5-bay-sata-sas-hard-drive-backplane-hd-036-bt.html).
You'd need either a bunch of eSATA->SATA backplates or a SAS/SATA multilane adapter board. You would also then need one or more corresponding add-in cards for your current PC plus appropriate cables.
Oh and a PSU jumper like this:
https://www.overclockers.co.uk/ek-water-blocks-ek-atx-bridging-plug-24-pin-wc-995-ek.html for the old power supply.
If your current PC
does have spare SATA ports and/or 5.25" bays, the cheapest option to address at least some of your drives might be to put a hard drive back plane into that and probably supplement
it with a SATA card (OC only have 2-port examples, you might need more ports:
https://www.overclockers.co.uk/star...press-controller-card-pexsat32-cc-004-sr.html).