Is it possible if you buy off the shelf from one of the mainstream box makers or it's only possible if you custom build one with fancy cooling system like water cooling or passive pipe cooling systems?
check out ANTEC cases and fans - they are quiet! Also google and ask around as to which harddrives are quit and still reliable - I have some old western digitals that whine, my Maxtors are quiet for example. consider placing the machine away from you as well.
A notebook computer with a large flash card instead of a disk should be almost noiseless. Put the big noisy computer in another room and use Remote Desktop Connection to access it.
I bought a Dell E520 (Intel Core 2 Duo) with basic (cheap) built-in graphics chip during a year-end sale. I added a second hard-drive and turned on the built-in hardware-based RAID function (probably costs about the same but I wanted two different HD manufacturers for my RAID array). I also added a multi-display PCIe card (NVS 440) that has no fan (a full card,large heat sink). In opening the PC to install my stuff (and probably voiding the warranty) I noted a very large heat sink on the CPU. You do not hear the PC running except on turn-on when the cooling fans starts up. They immediately slow as the magic on-board speed controller is activated. I'm sure the on-board video chip would produce less heat than the NVS 440 so I'd say the machine would remain "noiseless" without the NVS 440. The hard drives have a BIOS option for response speed vs noise. I've opted for the slower speed, lower noise (none). At this time my trading room is at 70 degrees F. In the summer it goes to 78F. Whether the fan(s) will speed up so that they become audible is to be determined. Jack