Can't you just use a faraday cage, sound matting, and add some filters on the mains power coming into the power supply and any USB/monitor cables coming out to decouple any potential crosstalk? This is a solved problem in very sensitive equipment like electron microscopes, where the smallest signal or vibration can leak (from a USB keyboard or another machine on the same power line on the other side of the building, for example) and effect the electron beam.

The other alternative would be to raise the electrical and auditory noise floor to the point where the leaked signals get lost in random noise. Then you'd have to perfectly synchronize the external sensors with the internal clocks to be able to extract any useful information.

Obviously these solutions aren't perfect (what is?) but short of placing a spatially filtered passive radar in the same room, you wouldn't get any usable information leakage.

You're still just raising costs, not making it impossible. Which is fine, because security is always about raising costs to make break-ins not worth the while.