You know, it depends on how long you want the back up power to last. If you are just trying to survive a short term power outage, say a week, your best bet is probably going to be an internal combustion engine powered generator. Get a good one that puts out a clean waveform. I assume there is a forum here on gensets. Best bet might be a small diesel one, just because you can avoid gasoline storage issues. Then again you could just buy some PRI Gas Treatment and Fuel Preservative (look it up; it even revitalizes stale gasoline) .
An argument can also be made for a duel fuel propane gasoline generator.
Point is that for simplicity and energy provision IN THE CURRENT ECONOMIC system, it is hard to beat a genset.
So, that is option 1 for short term electricity back up. And it all depends on what you define as short term.
Option 2 is to buy some deep cycle lead acid batteries, a basic battery charger, and an appropriately sized inverter. YOu will size the inverter according to your projected highest demand or load requirement. (Spend $30 bucks and get a P3 P4400 Kill A Watt Electricity Usage Monitor. Measure all your appliances usage. I have read that sometimes they can smoke, so be careful about leaving it hooked up to a large demand device for long term measuring. I know from using mine that I can run a full load of laundry in my old Kenmore washer for 1/5 of a kWh, believe it or not, because I measured.)
Anyway, you get about 700 charge discharge cycles out of lead acid tech if you don't discharge more than 50% in a cycle. I think it's something like 300 cycles if you discharge to 20%. Now if this is truly for back up and you don't have a lot of power outages, you could just assume you are going to discharge to 20%. That gives you about 800 Watt hours per cycle per battery, or 4kWh for a 5 unit battery bank. (A typical $100 deep cycle battery has about 1kWh of energy storage, but you MUST NOT use all of it. You will kill the plates or dramatically reduce the life of the battery.) Recall the examples in another member, Mike's excellent post. 4 kWh would be enough to run lighting, small electronics and such for multiple days. IF you want to run heat generating equipment or other high demand equipment you are only going to get hours of usage. (So don't. Get a wood stove, propane stove, gravity fed pellet stove, or kerosene stove, or a gross of candles, if you want heat back up. And heat water the old fashioned way for sponge baths. (Or buy more propane and get a small on demand water heater) Get some 5 gallon propane tanks, and conversion hoses for light and cooking using camping equipment. Don't forget extra mantles for the lamps.) Believe it or not, old school Hoover type vacuum cleaners actually pull 13 or more amps, rivalling electric heaters. So if you want to run a vacuum, you need an inverter that can handle that load (approximately 13 x 120 = 1560 watts. So minimally a 2000 Watt inverter. Now, 4 kW would only let you run that for about 2 and a half hours. But assuming you even need to vacuum during a power outage, how long will it take to do so? Point is that you could do it. Bottom line is that you need to size your battery bank and inverter for both maximum demand (+safety margin for initial peak demand) and the total amount of energy (kWh) you need to have during the outage.
Again, the point is that if it is a short duration outage you are planning for, you could conceivably do without the wind or solar or small scale hydro options, and just get batteries and an inverter. Btw, you're going to have a little money into the battery wiring harnesses and clamps and such as well, and a $25- $60 charger.
Now, if you are talking about longer duration outages, what you are really merging into is ALTERNATIVE or OFF GRID systems... Totally different animal and all the preceding posts address that, when it comes down to it. Basically, I am proposing that prepping for a power outage is a totally different situation from the kind of long term or permanent independence from the grid that one strives for with alternative energy.
So what do I have, or what have I done? Well I have multiple 3k to 5k gasoline gensets, an 8k diesel unit, and lots of fuel for both types of units. I also have, UNDEPLOYED, about 6kW of solar panels of different sizes, a number of inverters, multiple charge controllers and a dozen or so lead acid deep cycle units. I figure I will eventually move to lithium or whatever once that tech further matures. I figure in a true grid down SHTF, TEOTWAWKi, Mad Max, World Without Law (etc., etc.) situation, those older batteries still have value when paired with a single panel, small charge controller, and small inverter (e.g., 1kW). Would make a huge difference for basic lighting, electronics, and water pumping. Could barter those for the goats, chickens, etc. that I don't currently have, for example. Again, assuming I had by then switched to Lithium. But I have to admit, I might not switch to Lithium. May wait to see what "Donut" comes up with or Samsung... The point is that I don't plan to even deploy solar until I actually need it. Power in Southern Coastal Oregon is relatively cheap now, and given a normal winter, I wouldn't get much solar production anyway.
I hope this was all clear. Bottom line is that again, for short duration - even a few weeks - you are probably better with a genset and fuel... Note that part of my reluctance to deploy solar is that we are overdue for a giant earthquake and I don't ven know if my house will survive it. (Or me.) So I figure better to store panels and preps in fireproof and quake proof storage and be able to deploy if I really need it.
Also, my numbers are approximate. Different deep cycle batteries, for example, will have different energy storage capacities and tolerance for discharge.