It is entirely possible - in fact i have built a couple of PCs this way - however it does require some careful consideration.
1st step is to eliminate the heat being generated in the first place - ie pick low power components, undervolt your processor etc.
A64s and Semprons are fantastic for this - undervolted they easily compare to laptop processors like the pentium-m and turion chips, yet you will still be able to run at stock speed or faster!
2nd step. Consider what else you will hear if you don't have all those fans running. Hard drives make horrible high pitch noises that are often more annoying that the whoosh of air from a fan. Beyond the hard drive you may have annoying coil whine from the motherboard or PSU.
A quiet, suspended or enclosed hard drive is essential. Use either a samsung spinpoint or better yet a quiet laptop drive.
3rd step. Consider airflow carefully. A very effective single fan system would use a PSU with a slow 120mm fan like a seasonic S12 placed above a large CPU heatsink. This will draw heat away from the CPU, and from the case. If all your other components are relatively low power, this should happily cool the whole system. The silverstone SUGO1 case is ideal for this.
Don't forget the other critical components though. You will need to make sure the graphics card, chipset and critically the hard drive do not reach excessive temps. Most graphics cards and chipsets will run happily a lot hotter than you ever imagined (I've run many at over 90deg for years with no ill effects) but especially if you have important data on your hard drive, I would be very wary of regularly letting it exceed 45-50deg.
By closing off ventilation holes etc to direct just a little airflow over these critical components you should be able to achieve your goal of a PC with a single fan. Completely fanless is possible, but you will have to compromise performance. Water cooling IMO is not an option as the pump will be louder than a single quiet PSU fan and without any fans at all, certain components could still reach critical temps.
Marc
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