So from the looks of it, it seems that you kind of want a Media server / HTPC.
Something like Antec 300 or Anter 300 illusion would work fine for you on the case front. The illusion has 4 adjustable fans that will keep the HDD's cool and it should still be whisper quiet. It has 6 internal hard drive locations, so that should be easily enough for storage. 6x2TB in raid 5 = 10TB And you will probably do fine with less.
A dual core CPU should be enough juice to run your CPU stuff such as i3 2100. However looking at reviews the integrated HD2000 does have some issue with 1080p playback with lots of stuff on the screen so I would recommend a GPU. Something in the $100 range max should be able to handle things great.
Probably a motherboard which supports your desired RAID solution would be fine. If you want to go overboard a raid card with support for 4 or 8 HDD's would of course be great, but it's pretty hard to justify the $300-500 price tag.
Pretty much the standard 4-8 gigs of ram.
5400 or 5900 RPM Green drives are generally the way to go for these servers as with raid they are fast enough and they make less noise and run cooler.
They are also dirt cheap. (~$65 for 2TB)
For a tuner card,
Ceton InfiniTV 4 Quad-tuner Card $299.99
Is a bit costly but it has 4 tuners so it is worth a look, it should be great.
If you want to play blu-ray's get a blu-ray reader, otherwise stick with a DVD writer.
PSU: They are quite expensive, but I would look at PSU's from seasonic. 2 fanless models FL-400 and FL-460 should work just fine for you and have enough juice for your things. If you want to get one with fan the X560 would easilly have enough power for your needs and most likely it would not run the fan at your power usage anyway.
All 80+ Gold rated.