What to use to lower the computer noise in a wooden cabinet?
If most of the noise is coming from the hard drives, rather than the fans, then you need to dampen the vibrations from transferring from the hard drive enclosure to the furniture. 

The vibrations could be made even louder by the fact that it's on a shelf supported on its ends only - this could cause the sound to be even louder if the hard drives vibrate at a frequency that lines up with the normal mode of the shelf. 

I would recommend the following:


See if the noise gets better by adding a support to the shelf that isn't just the ends. Try holding it with your hand, or propping it up with a board to see if that helps reduce the sound. If it does, look at adding support to the shelf.
Add some velcro to each of the hard drive sleds
Use rubber padded screws to attach the hard drive to the sled
Put a rubber pad underneath the hard drive enclosure.


It may take some trial and error to figure out what helps and what doesn't, but the basic idea is to attenuate the transfer of the vibrations.

Or, you could just switch the drives to SSDs, or select drives that include built-in noise attenuation.