Most likely, an older computer would work perfectly fine (I used a Pentium 3 866mhz as a web/file/print server at home for a year without any problems, and I've had a P2 233mhz set up for temporary access). Only reason I could see building a new one would be if you didn't have an older one or if you wanted a massive amount of storage.
If you do decide to build a new one, though, it doesn't make all that much sense to "just" make it serve files. Why not give it a little bit more juice and let it act like a media center to play back video and music and the like?
Motherboard: EVGA 113-M2-E113 AM2+/AM2 NVIDIA GeForce 8200 HDMI ATX AMD Motherboard
- $90, $70 after MIR
CPU: AMD Athlon 64 X2 5050e Brisbane 2.6GHz
Memory: Kingston 2gb DDR2 800
Optical Drive: LITE-ON DVD-Burner
Power Supply: FSP Group ATX300-PA 300W ATX 12V
Case: Rosewill R222-P-BK Black Steel ATX Mid Tower Computer Case
Total (before hard drives): $242 before rebate
If you really just want it to be a server, the above will most likely be overpowered, so then I would recommend switching out these parts:
Motherboard: Asrock A780FullHD
CPU: AMD Sempron LE-1200 2.1ghz
New total (before hard drives): $179
Both the motherboards will work with either processor, so if you want some more power you can always get the faster dual core processor in the second build. I'd say go with AMD because (a) they're slightly cheaper and (b) if you're just serving files, or even just playing back things, either will be more than powerful enough.
Hard drives are dirt cheap now (at least in $ per gigabyte) - they have terabyte drives going for about $100 and smaller ones for even less, so just add how much storage you want. You'll need to get a SATA cable
for each drive and a Molex -> SATA power adaptor
for every two drives, but each of those are fairly inexpensive.