Thank you all for the replies.
The 10GbE option I had in mind at first was supposed to work like this - the server and switches would support 10GbE speeds, while each client could stay on its 1GbE integrated NIC, which would still yield the theoretical maximum of 125MB/s per PC. However, I didn't find any reasonably priced switch with 10GbE ports in my vicinity, so I won't be pursuing this option further.
I have heard about FreeNAS and deploying it on a self-built server; however I don't know how important the CPU/RAM are for the server itself, as it should only work as an 'offline dropbox' for working file, and I don't know how much babysitting a FreeNAS server requires once it's built and turned on. I've used a few linux distros before, so yes, once they're configured they are set; but when something goes awry it can be a pain to fix.
So my latest thoughts are on a 4-bay Synology DS416play, which has 2 GLAN ports and with proper switches that too support link aggregation (like TP-LINK TL-SG2008) this could be a good balance between price, setup ease & servicing.