it shouldn't hurt anything to try running that all off a battery , but the problem may be how long can you run before you drain your battery . if you have to buy a power supply , it probable isn't worth doing . now if 1 battery doesn't run long enough , you can add another and keep in mind that a battery that won't start a car in January in Minnesota is probably still good enough to suit your purpose . i used to use a has-been indash and some 6x9's for a stereo in my garage and it worked out ok , i would have to charge my battery now and again .
another thought is , that supposedly a computer power supply has a 12 volt dc output , with enough current to run a small to medium subwoofer set up, i don't know all the details or if it is true , but i wouldn't be surprised to find out it works . i have seen many computers on the curb , free for the taking . people even pay to dispose of them