There are special brackets for mounting HDDs, where there are rubber rings interposed between the mounting screws and the bracket, in order to prevent the transmission of the HDD vibrations to the chassis, which usually greatly amplifies the HDD noise.

With such brackets the noise is usually reduced a lot. There are some computer cases that are advertised as silent cases and which include such brackets by default (e.g. from Fractal Design).

Someone did, but they're still slightly more expensive: https://en.wikipedia.org/wiki/Solid-state_drive

I found that putting them on something soft helps a lot. It seems to be mostly vibrations that are transferred to whatever it's sitting on.

It's probably coupling the vibration into whatever shelf it's sitting on, which then transduces it to the air. Set the unit on some foam, it helps a lot.

In your case I would probably buy a couple of floral or similar foam bricks and try to decouple device from whatever surface it is standing on.