I do have an optical drive and the GPU is an older GTX 750 TI, which doesn’t get much use and I don’t think the fans ever kick in. Definitely not when I want it to be quiet anyway! There’s also a couple of mechanical HDs.
I’ll try the p/s, CPU heatsink and rubber screws and take it from there.
That CPU... https://ark.intel.com/content/www/us/en/ark/products/88184/intel-core-i5-6500-processor-6m-cache-up-to-3-60-ghz.html
It puts out 65W of heat and it's an LGA1151 socket.
You can get something like this for it:
https://www.quietpc.com/mugen-5-pcgh
Which will take the CPU fan down to 15 dBA at peak.
There's no mention of a GPU, so I'm guessing you're using the onboard on the motherboard https://www.gigabyte.com/Motherboard/GA-Z170-HD3P-rev-10
I think you should be fine to get a big case that could take a larger CPU heatsink and achieve total silence.
The only other fan in your system appears to be the PSU:
https://www.evga.com/products/product.aspx?pn=120-G1-0650-RX
Something like this: https://www.quietpc.com/fsp-hyper-m?product=5375
This is only 600W, but without an optical drive or GPU... you don't need 650W.
With a new CPU heatsink and a new PSU and a case, the entire system would be under 20dBA (basically virtually unnoticeable).