SPCR

Is there an HTPC case with more than 4 HDD drive bays and included "silent" fans?
Of course there wouldn't be any GPU in it, just basic CPU with integrated graphics and minimal PSU, so hopefully that would greatly contribute to noise reduction.

by HTPC cases I mean cases that are shaped like this-

If you were to accept the not full shape as an AV HTPC case, i would suggest Fractal Design Node 304 Mini-ITX Case, can hold 6x 3.5 hdds.

If you do want a HTPC like AV design, then look into SilverStone Grandia GD07 & GD08 Media Center Cases.