Hi, if you are looking for super quiet and prepared to make some compromises then you can get a very quiet PC indeed.
Start with integrated graphics and then add a low end dual core CPU, Intel or AMD have viable choices. Under-volt it as far as possible with out compromising stability, maybe even under-clock as well if performance not desperately needed.
Add a large heatsink, eg Ninja or HR-01, and a 500rpm 120mm fan either on the heatsink and ducted to the case or on the case and ducted to the heatsink.
Power with a PicoPSU or other voltage converter type PSu that runs off a (fanless) DC brick.
Finish with a SSD drive. MikeC wrote about a PC of this sort they have in the lab and commented that it was virtually inaudible even in the anechoic chamber! Not sure if was an article or just a forum post on it.
EDIT: Mike's single moving part PC.
The Seasonic X series PSU runs fanless to higher power that a DC brick based once can and then very quietly beyond that, it isnâ€™t cheap though.
If an SSD is too expensive or small capacity look at a 5400rpm drive and put it in a Scythe Quiet drive or the like. (beware of some enclosures that donâ€™t deaden noise much though) Capacity maybe substituted by gigabit Ethernet and a file server / other PC / etc.
These are quite extreme examples, you could pick parts from the recommended list and hopefully end up with a PC with 2-3 120mm fans and as long as they are descent quality and run under 800rpm (preferable under 600rpm) it should be barely audible when on the floor next to your desk. If you want it on the desk you have to work harder to make it in-audible!
Definitely avoid having a hard drive hard mounted under any circumstances. Also try keep HDDs down to one.
Good luck, Seb