I recently acquired a free Devilbiss oiless "5hp" that is LOUD. I put together a little shed for it from scraps a few weeks ago. It's 3'x3' and the compressor only raises the temperature a few degrees when it's running (although I haven't run it continuously in real hot temperatures yet, only having it kick on a few times in 20 minutes or so). I did put
this 50 cfm bathroom fan in the "attic" and the exhaust blows through passages in the 2x4s that I cut out with a hole saw. It's plugged in to a christmas light switch that's in the garage, and I haven't even been using it. The passages split and recombine the exhaust stream around bends to disrupt the flow before it makes its way to the vent on the opposite side. I did a similar setup for the intake through the wall. I didn't insulate anything at all, but it is double-walled osb to make it more rigid. If you put your hand on the outside while the compressor is running, you can't feel any vibration. This double-wall setup is significantly quieter than the ones I've made in the past that were single wall with blue board styrofoam. Might just be due to being sealed up better, but either way it was cheaper and easier. I did put 1" of Styrofoam under the EPDM but above the attic airspace because it is black and I figured it would drop the temps from the sun.
You can stand right next to it while the compressor is running now and have a conversation at a normal volume. It isn't any louder than the house's AC unit right next to it. It's about the same low pitch as the AC also, which is a lot more bearable than the high pitched noise it was before. If you're inside the house on the opposite side of the wall, you actually can't even hear it unless it's silent and you listen for it. It's right up against the headboard of my bed, and it's quieter than the fan in the adjacent bathroom.