Damping material on the case doesn't cut down on sound escaping through the holes in the case but it helps stop the case vibrating.
I've been building my own very quiet PCs for many years now. It requires careful selection of components, always with a view to fitting big, effective heatsinks and as few, quiet fans as possible operating on the lowest voltage possible. My current PC is no louder than background measured in a typical quiet room - around 26-30dBA at 1 metre. For recording purposes any PC noise that finds its way into a sensibly positioned mic is lost in the general background.
This inevitably costs more than an off-the-shelf PC from HP, Dell etc. and means getting familiar with PC internals, their noise levels and their power consumption. Once built a quiet PC absolutely has to be stress-tested using software to send the cpu and graphics card to 100% and holding them there for several minutes while monitoring internal temperatures to check the cooling system can cope.
The PC in my sig has only three 140mm fans running slowly at 5 volts. One intake, one output and one in the cpu cooler, no fan on the graphics card or psu. It can hold 100% of cpu pretty much indefinitely so long as I clear the vents and coolers with air duster once in a while. The loudest components are the HDDs.
Just because a manufacturer claims their cooler/fan is super-quiet doesn't mean it actually is, at least not by audio standards. I suggest
www.silentpcreview.com is a good place to start research. Many if the articles are old, but the forums are very useful.