You can, but it probably won’t give the desired effect.
The reason is that car batteries don’t hold the amount of energy you probably want. Running nothing but one refrigerator off of a car battery for a few minutes may be enough to cause damage to the battery.
To run regular household appliances off of a car battery you must use an inverter with the appropriate AC voltage output. Once you’ve got the right voltage you still need to have enough watt-hours (total energy in other words) to accomplish the task ahead of you.
One sensible use of this principle might be to run a gas-fired heater or oven. The amount of electric power needed by these appliances is minimal because they get almost all their energy from gas.
In reality, many heaters have a powerful electric pump or fan designed to circulate the beach throughout the house; those particular electric devices won’t run long at all on a standard car battery.
And if you just wanted to run a couple of household LED bulbs for a limited time, again, no big deal for a car battery.
Anything designed to produce a lot of heat (or cooling) from electricity is probably unworkable.
Another consideration is that it is likely more trouble than it’s worth to wire a battery + inverter into your entire home wiring system. To do this safely you need to disconnect your internal home wiring from the grid power system.
It’s not safe to allow power to flow from within your home back to the grid during an outage because you don’t have control over where that power goes (possibly into a downed wire, or harming an electrical worked who was looking to fix whatever damage caused the outage)
Ordinary car batteries are not designed to be deeply discharged and recharged many times. They won’t last very long at all used this way.
But you can buy batteries that are VERY similar and put out the same voltages, etc, which are referred to as DEEP CYCLE or MARINE batteries or GOLF CART batteries, etc.
These cost maybe twice to three times as much, but they will last a hundred times longer than ordinary car batteries using them to run lights or a computer, etc, in your home.
But you still must be careful not to discharge them TOO DEEPLY. That will shorten their life drastically.