Freezer/fridge won't cut it
Your average fridge or even a deepfreezer simply won't cut it, I'm afraid.
The compressors used in such household appliances (even the larger ones) are built to tolerate a certain duty-cycle, and as such they will likely burn out VERY quickly, if run continously.
Even if you find a fridge or freezer with a compressor able to handle continous running, consider the fact that your fridge is quite efficiently insulated, and as such doesn't need to have very much heat-energy removed to maintain it's low temperature.
This means that the power-rating, and thus the heat-removal capacity, of even a very large freezer (say one used in a restaurant or grocery-store) won't be able to remove the large amount of heat generated by your typical performance PC. These days this will easy be as much as 300-500 Watt (thermal).
This is why some companies are able to sell refrigeration units for PCs at such high prices. These have been designed specifically for PC, and are able to handle the large heat-removal requirements as well as the need for continous operation.
Linkage:
http://www.asetek.com/
This Danish company produce high-performance units (product name: VapoChill) able to cool your CPU (but typically nothing else) down to below (or at least close to) freezing temperatures. This, however requires special care to avoid condensation on the chip and its surroundings.
http://www.alphacool.de/
This German company produce a series of VERY nice "water chillers" (product name: Titan). These work by using refrigeration techniques like the above VapoChill product, but instead of cooling the chip directly, they cool water instead, as well as providing the reservoir and circulation-pump. Such a unit will work as a VERY efficient external radiator/reservoir/pump, and even comes with electronic temperature, pump and fan control, able to maintain a given water temperature. Even though the larger models could probably keep all the (water-cooled) components in a PC at very low, if not exactly freezing temperatures (say 5-10 deg.C.) their real advantage, as compared to the above direct-refrigeration units, is that you can set the thermostat to maintain a water temperature just around the ambient air (say 18-22 deg.C.) and thus avoid any condensation trouble in your PC. Another advantage of this particular product is that they appear to have quite strong pumps, thus it may well be possible to place the cooling-unit in another room (or even outside, if sheltered from the weather), to avoid excessive noise and perhaps increase the cooling efficiency. A final, less obvious, advantage of the Titan series of "Water Chillers" is the possibility of mounting a conventional, small PC radiator with a low-noise 120mm fan INSIDE the PC case, and use the refrigerated water to cool the air in the case. This way you could simply circulate the same air in the PC case, completely preventing dust and smoke etc. from getting to your hardware AND you won't get any condensation problems, even if you set a very low water temperature (since the case air will be cooled as well), at least not inside the case (you might, in an extreme case, get condensation on the outside though, but assuming your case doesn't rust, wouldn't it just be cool to have a moist PC!!!)
Both of these products are VERY expensive though, and will use quite a lot of electricity. So I would still recommend a conventional water-cooling solution for most PC users, with a passive or active radiator depending on your noise-sensitivity and cooling-requirements.
Also keep in mind that the size of your radiator, especially if passive, is the main factor deciding the heat-removal capacity. Thus you might get quite a good result simply by finding an old (or new) household radiator of suitable size, mount it on the wall near your PC and use it like any other, dedicated PC-radiator, to cool the water for your PC. If you live in a cold climate, mounting the radiator outside will of course increase its efficiency (but don't forget to add anti-freeze to the coolingwater, unless you NEVER switch off your PC!!!)