It depends what you want to do really. Is it just going to be two IDE drives? Or will it be more later?
My guess is that it will be more later. In that case, you could do like you said and setup some crap cpu/mobo system and put linux on it. A short list of recommendations might include:
-Make sure it has a number of SATA ports on the board. The more you get on the mobo, the less you'll have to add in with cards later.
-Get the slowest CPU of the line or get a mobile CPU and use Q-Fan control in the mobo BIOS to turn down the fan speed when it isn't doing any work. E.g. an Athlon 64 3000+ or similar. If you put in the research, you could learn how to make linux scale back the cpu speed when it is not in use to save power/fan noise.
-Do a little research and learn how to spin down the disks when they aren't in use to save more power and wear on the disks.
-Use linux Logical Volume Management (LVM) so you can expand the capacity of your disk array without having to take everything off and reformat or other annoying hacks.
-Think about RAID so you don't lose all of your hard-won media in a single disk failure.
-If you want to get super crunchy, investigate wake-on-lan so you can turn the box off, but wake it up remotely when you want to access something on it.
If these sorts of things are out of your ken and you don't want to devote the 10's of hours it may take to learn how to set things up, then take the advice of other posters and purchase a slightly more expensive ($300+) NAS. I don't have any product recommendations for you there, unfortunately.