Is it possible to use a power inverter to run a sump pump?
Sizing the Inverter

If you want to run an appliance like a sump pump from an inverter, you need to make sure that you get one that's large enough.

The pump will be rated according to how many amps it draws (probably in the range of 5 or 6, based on a quick internet check). However, when an AC motor starts it uses a much higher current, maybe 2x as much. Ideally you would measure this surge current amount and duration so that you know what you're working with, but you could assume double the current for several seconds. So if you have a 5 amp pump, lets assume its surge current is 10 amps.

Inverters are usually rated by watts, but you can compute the amps by dividing by 120V. Inverters usually have a continuous wattage rating and a higher surge rating. Unfortunately, cheaper inverters will only surge for a fraction of a second before cutting out, so you can't just assume that the surge rating of the inverter will be able to supply the surge current of the pump.

The only way to really be safe is to get an inverter whose continuous rating is greater than or equal to the surge current draw of the pump. So in this case yo'd be looking at a 1200 watt inverter (which may have a higher surge rating).
[Some of the nicer brands of inverters will publish how long the surge rating can be used for, e.g. 0.5 seconds, 2 seconds, 60 seconds. If you know that your inverter has a long surge time, you may be able to get a smaller inverter].

Sizing the Battery

If this is just for a one-off emergency usage you may be able to run the inverter from a car battery, but you should at least have a sense of what you're getting into. To compute the battery runtime you need to know the amp draw on the battery. Note that the amps coming out of the battery will be at least 10x higher than the pump's amps, due to the much lower voltage. A 5 amp / 120V pump is using 600 watts, which is 50 amps at 12V. Assuming an inverter efficiency of 80% you're pulling around 62 amps from the battery. The capacity of car batteries varies based on model, but a good sized battery probably has 50 amp-hours of capacity (that's 50 amps for one hour, or 25 amps for 2 hours, or 10 amps for 5 hours). So you're looking at getting about 45 minutes of runtime until the battery is totally dead. (Of course it's not good to run a car battery all the way down, and then you won't be able to start your car, so it probably wouldn't be prudent to actually run it for 45 minutes).

If this is going to be a semi-regular event, you probably want to get a marine / deep cycle battery or two. Those have a higher capacity and will withstand discharges better.

Other Considerations

Automatic Transfer: Most small inverters just have regular 120V outlets that you plug your load into. If you want the inverter to be hooked up full-time and transfer from mains power to battery power, that will require something more sophisticated and probably a professional installation.

Wire Size &amp; Fuses: As you can tell from my rough example numbers, you're looking at 60+ amps from the battery, which is a LOT. Standard household romex / extension cords is not going to be safe for that amount of current. You must properly size the wire or you will have a fire.

In addition, you should have a fuse on your wire with a rating less than or equal to the wire capacity. The purpose of the fuse is to blow before something catches on fire. If you're regularly blowing fuses it means there's a dangerous problem with your setup.

DC Pumps: If you want to run a sump pump from a battery on a regular basis, you're probably better off just getting a 12V DC pump and avoiding the inverter. You can run those directly from a car battery.