There are quite a few requirements in there so I will break it down a bit:
A silent, even a no moving parts silent, computer without huge 3D graphic needs can be done. The best way to do it is in a case that acts as a CPU cooler like the ones in the link to computeraudiphiles above. Do you need one with an optical drive to rip your CDs?
How much space do you need?
An uncompressed CD is 650MB, loss less compression cuts the size in half. Examples what 80€ buys you in storage space (with loss less compression): 120GB SSD -> 360 Albums (You will want to install your operating system and software here, so it's more like 100GB -> 300Albums), 1TB notebook drive -> 3000 Albums, 2TB desktop drive -> 6000 Albums.
Speed is not important for music storage. The speed of all drives exceeds 50MB/s in all cases, the bitrate of uncompressed CD quality audio is ~0.17 MB/s.
SSDs have no moving parts and are completely silent, but they are expensive.
Hard drives have moving parts, 3.5" drives are louder and cheaper, 2.5" notebook drives are slower and less noisy (in general).
How do you want to connect the drive(s)?
Internally (SATA) you will want only a SSD, you will definitely hear a 3.5" drive and a 2.5" drive could be audible over low volume music.
External connection (USB, Firewire, eSATA) is easy and the drives are portable, but the cables have a very limited length (1m-1,5m). You would likely still hear the drives.
Networking would be my choice. Either run a network cable to where you put the drives or use wireless. You need an external enclosure (NAS). You can put it far away from your listening station so you don't hear it and use the larger, cheaper 3.5" drives. Synology and Qnap both make good home NAS systems. The Synology DS213 Air
is the only one I could find that has built in wireless. It is a good place to start, it holds two 3.5" drives and has a low power consumption (8W-18W depending on activity and drives). All you need is an electric outlet, you can put it almost anywhere.
How do you want to connect your DAC?
I don't have audiophile grade equipment. The onboard sound on my last mainboard was terrible so I bought a sound card and I could hear the difference. My surround system has an integrated DAC and connects via Toslink. This setup is simple and not electrically connected.
A good USB connection to the DAC is superior because it has zero jitter. Excess jitter can cause audible distortions, low jitter is inaudible. "Low" is not precise and it depends on the rest of your equipment and your ears.
USB is superior, but I wonder if it's the kind of audiophile obsessions that benefits dogs and bats only
. Poeple on the audiophile forums might disagree.
I would start with SP/DIF and if you can hear distortions upgrade to the high grade USB equipment outlined in audiophile's system build.