It takes quite awhile for a large volume of water to cool off. Usually as long as the temp stays above 72 you will not have any major problems. Of course, if you loose power for several days, this becomes a big problem. Also, heaters draw a lot of power when they are on, so they would deplete a battery quickly.
Lighting is something the system can do without for a fairly long period. Even a week or so without light usually isn't a big problem.
Circulation is the key item. For a few hours the tank can get by, but after than you need to do something. Battery operated air pumps are a good low cost choice, but you will need to change batteries after about 12 hours at best. A standby generator would be ideal, but it's very expensive.
Battery backup power supplies like you get for a personal computer, might run a powerhead, but they usually will not last long in that mode. The are designed to provide a few min of power to normally shut down a computer, not power something for hours.