Fan noise is mainly generated by the tips of the "propeller" fan blade hitting the air and accelerating it to generate the airflow.  A few years ago when I spec'ed out my new editing computer, I went for a minimum size of 120 mm fans (three of them inside the case), which was about the largest diameter readily available in stores at that time.  Today there are larger fans (140 mm to 200 mm) which rotate at slower speeds for a given air flow, as well as centrifugal fans available.

Also to consider is the fact that a "propeller" fan will have the highest turbulence (read: noise generation) at the tips where the velocity is the highest, while a centrifugal (squirrel cage) fan accelerates the air flow from the centre and "throws" it out through the blades, generating much less noise.  Internal small centrifugal fans to exhaust the warm air do exist, but may not be very readily available.  One work-around would be to get a centrifugal fan that is mounted externally and blows the air into the case.  Of course, you would have to ensure that the air path from the entry to the heat-generating components is as open as practicable.  The solution using large or centrifugal fans should be more cost effective than a more exotic fanless solution, and it would allow the computer to be located under a desk to additionally attenuate any remaining fan noise.