I would suggest a P(something).
A file server doesn't need a whole lot of processing power, but *some* is required. If you go with a 386 the bus speed on your motherboard is going to be slow. It's been a long time since I touched one of these, but if I recall PCI slots didn't start coming out until 486's?
You might find you hit problems with large drives aswell - your motherboard is unlikely to recognise anything above (thinks for a moment..) 8GB on a 386. You might have problems finding a BIOS upgrade that'll allow larger hard drive sizes too..
All in all, a 386 is likely to cause you more problems than be worth it. If I were you I'd lay out the $50 or so for a simple P2 or something and go from there.
Once you *do* get it set up, recognising drives, you can get samba to share them out and then.. yes, just map a drive on your local client to the networked drive and it'll look like a drive on your local machine.
One more thing - investment in a 100Mb card / switch is *definately* worthwhile when doing this - transferring large files (hundreds of megs) over a 10mb shared hub is painful when compared to a 100mb switch.