Don't see why it wouldn't work with a NAS running a CIFS share.
Yeah after thinking about it a bit, I don't think it needs to do anything other than provide access to the .data and .bin files. Right now the computer that is acting as a 'server' is really just functioning as an interface between the drives and the network. Windows is running on the C drive and a separate pair of drives houses the MS Access .data and .bin files, so I think really the rest of the computer or even what OS it's running, whether Access is installed etc. is irrelevant.
I think just a very simple NAS enclosure should do the job.