I find this thread title silly when even a gaming PC with proper fan selection and fan curves is still gonna be quieter than any air conditioning.
A quiet fan-cooled system will not be detectable in a room with a typical ambient noise floor. To those who say otherwise, I'd say you should back it up with a blind test
Like many others no aircon here. The problem is what you individually are aware of and annoyed by vs. what becomes an accepted part of 'normal'. It's a bit like vinyl surface noise, active monitor hiss, or gaps between tracks that should transition seamlessly - pretty much everyone can identify them once it's pointed out, but for some people they're mentally filtered as 'normal' when not concentrating on them, while for others they're a disturbance that shouldn't be there, and that becomes increasingly annoying. If you don't notice it then consider yourself lucky. Another example in a different field - a graphic designer I used to work with got annoyed by font details I could identify when he pointed them out, but wasn't generally aware of, let alone bothered by. Some here have a similar reaction, usually with mismatched fonts on displays and front panels, or with the allignment of knobs, sockets etc.
For me the annoyance is in an identifiable part of the background noise that I can do something about. The character of the noise matters here, not just the level, which doesn't show in the dB noise rating most fan and HDD manufacturers provide. White or pink noise will be less identifiable at the same level than specific frequencies - hum/buzz vs. hiss. Another aspect is whether it's constant or variable - a step transition will be clear whereas a slow transition may go unnoticed. As for being able to do something about it, if it needs a fan then it needs a fan, and it won't be annoying if I've used the least intrusive fan practicable. Like I said, it's a weird personal thing.
The one that I've put particular effort into is the HTPC in the living room that runs mythtv, so powers up to record things then powers down when it's done - assuming I'm not watching something on it. That state change draws attention to it, especially if I'm doing something quiet like reading late at night (so lower background noise). It's effectively a blind test in that I'm not looking at it while I'm reading, and not aware enough of the time or the recording schedule to know when it'll power up or down. It's now fanless with SSD so nominally no moving parts, and I'm hearing the change in noise that's presumably from the PSU coils - oddly the 12V power brick is noisier when it's in standby, and sometimes has me wonder whether another one would be quieter. There's also an audible change when the TV changes standby state - something I didn't know it did until I was trying to work out what the noise change was caused by and stuck a power meter on it.