A PII should be fine for a small fileserver. Add as much RAM as you can to it, and do a few tweeks to use it, and you'll be good to go.
But, heck... I ran a P1/133 as a fileserver for about 3 years. It would bog a bit on the integrated virus scanning I installed, but other than that, it never really complained.
I currently have an old recycled Slot A Athlon 650 running as my file server, NAT router, firewall/packet filter, FTP server, HTTP server, SSH server, virus scanner, and still have enough processor left over to run a game server or UAE with a CNet BBS on it when I feel like it! (Note, I don't really recommend loading that much crap onto one box, but even as such, it gets surprisingly good uptime. Currently on about 2 weeks. Not bad for a junk box with generic cheapo everything.)
I had originally thought about using an old Mac I had sitting around, as well. But also, it was SCSI and I didn't want to spend that much on disks. Plus I wanted the ability to add a 2nd NIC for NAT, Firewall, and packet filtering. While the Mac did have PCI slots, they weren't very compatible and I didn't have drivers for a 2nd NIC. :/ Hence, I shelved it. I think it's now in the back of my attic, somewhere.
Anyhow, I agree that I wouldn't use the Amiga for this. An 060 Amiga is valuable, rather slow (but fun) and somewhat rare; while an old PC is cheap, fast (but boring), and easy to replace parts on.
