Well, I can play it decently on Standard maps on my Pentium II 450 MHz desktop with 128 MB RAM, although it does run a bit slowly. Anything with a 1 GHz processor should be more than enough. I'd also look for 512 MB RAM if you want to play Huge maps.
For free hard drive space you'll need about 1.5 gig for the game itself. CivIII mods tend to be really big in size, although it varies a lot. Rhye's of Civilization is 1.5 gig if I remember correctly, so you might want as much as 5 gig just for mods/saves. Also make sure you have a way to use broadband Internet - you don't want to download a 1.5 gig mod on dial-up, even if it is compressed.
The manual also specifies a 4X CD-ROM drive as required, although since the game runs off the hard drive you might be able to get by with a slower one. And finally, there is a 1024x768 minimum resolution requirement that is entirely necessary (unless you want really strange scrolling).
Note that you can run the game with much lower specs - even Huge ought to run decently with a bit lower specs - but these are what I'd recommend if you want to be able to play any type of CivIII game without having to worry if it'll run well.
