Completely passive. Possible, why not? Expensive really needed very rarely. A couple of good fans ~500rpm to hear lot of hard work.
• Pick up is not much basking motherboard with a certain socket.
Heating modern motherboard can be neglected. Big size, little heating components. The South bridge does not emit much heat, the food chain - is still limited by the heat dissipation from a CPU, the large heat dissipation will not be there.
• With this socket you need to find (if one exists) processor which does not require active cooling.
Any, for which you will find sufficient exhaust heat of the cooler. And body. The case should not impede free air circulation. Top radiators cope even with the i7.
• Video card should not require active cooling.
Interesting, in terms of software, a more modern requirements. This Is Microsoft Office. Browsers, no matter how be surprised, but some sites will not fully open due to its great functionality.
What for you video card?
The most important thing is the power supply
Yes, this is the most difficult item in the first approximation. Namely passive blocks of the ATX can be counted on the fingers: https://forums.overclockers.ru/viewtopic.php?t=422392
But the second question is: why limit it to ATX? pico-PSU in the 150W is enough to power the system without a video card for the eyes. Processors with a modest TDP - hundreds of watts.
And the third point: if the PSU is cooler, it does not mean that he is rotating. At such low load more weighty list of power supply operates in the passive mode, despite the physical presence of the cooler. The list on the same link.
Not completely passive. To hear a couple of high-quality coolers in 500rpm - it must be out of town. Not in the city. To cool the ardor of any desktop 1-2 CPU coolers at 500rpm - no problem.
Much more expensive, but in principle any configuration, including a noticeable acceleration to gather ITS. (what to collect, the factory closed ITS garbage)
Well, after toying with a love triangle budget-performance-size