Update: Added info about Zeolite-water ice makers and re-edited.
When your electricity fails, with it goes your ability to refrigerate. A well-insulated, full refrigerator will keep the perishable contents for perhaps 4 hours with the door kept shut. A full freezer should be fine for 48 hours. See Keeping Food Safe During An Emergency for further recommendations.
So, what can you do if power goes off for days or weeks? Well, you could use a propane refrigerator or if you would prefer to prepare ahead of time and like the notion of both independence and sustainability, you could try building a solar ice maker (pdf).
Solar ice makers seem like a fairly robust technology. The only moving parts are valves, no horrendously expensive or exotic materials are needed, and so long as care is used in building the device, it should function perfectly well with little or no maintenance. The only scary part is the fact that the most common variety uses calcium chloride and ammonia as the generating medium. Care is needed during construction.
Research is also being done on solar ice makers that use methanol (pdf) and activated carbon. Unfortunately, methanol is toxic, and I have found no information on how to make it (unlike activated charcoal). There is reference (pdf) to using ethanol instead, which can be made at home. Ethanol, however, is corrosive to most metals - your ice maker will be eaten from the inside out and eventually fail.
Yet another kind of solar ice maker uses Zeolite and water as the working materials. Zeolite is a mineral, available in natural or synthetic form. Water is free, available just about everywhere, and is much safer and easier to use than ammonia. For the combination of system safety and availability of materials, a zeolite-water ice maker appears to be an excellent long-term ice making solution.
Unlike electric or propane refrigerators, solar ice makers don’t continually refrigerate. Solar energy is used to drive the fluid (ammonia, alcohol, or water) out of a material (calcium chloride, activated carbon, or zeolite). When the material is “dry”
, the fluid is reabsorbed by the material. This process cools the liquid to a point where ice can be made. In short, solar energy is used during the day, and ice is made at night (generally). While collecting your ice in the morning every day may not be convenient, having enough ice to keep perishable items cold just might be worth the price of admission.