A fanless desktop is actually easier than ever, thanks to Intel and AMD dropping 35W chips left and right :)
On the Intel side, a 35W i3/i5, a Thin Mini-ITX and Akasa's Euler heat sink case, and you're good to go!
On the AMD side, get the Athlon 200GE (or Ryzen 3 2200GE / Ryzen 5 2400GE if you...