Well at some point you simply have to deal with a huge bundle of wires plugged into lots of terminals at the Arduino end. In case you're using a UNO/Mega or similar with sockets on the board, you can get connectors and crimp pins that you can use to create headers that plug in quite neatly, and these guys
have some suggestions about that. You will probably want to use a break-out shield of some sort though.
I don't know how many places you have at the table or how many wires per place, but you can get ribbon cable with up to 40 conductors in various lengths. If you use a ribbon cable per place, I imagine that would keep your cable routing round the table nice and tidy. Depending how many wires you need per place, you may be able to use something more compact such as CAT5 cable.