I think the answer to your question boils down to recognizing that: 1 watt (AC) = 1 watt (DC) = 1 volt X 1 amp
Find out how much amperage your sump pump draws.
Multiply that by 120 volts to get the wattage of your pump.
The amperage rating of your pump is "real" power, but your sump pump get's it's power from an electric motor, which has a high inductive reactance, meaning that a lot of the power just goes into creating magnetic fields (which doesn't actually pump any water). So, boost the wattage of your pump by a guestimated 35% to account for power factor.
Multiply that by 24 hours to know how many Watt-hours you need to run your pump off batteries.
Multiply that number by two because you don't want to drain your batteries down by more than 50%.
If you're using 12 volt deep cycle batteries, then divide the above number by 12 to get the amp-hours of battery life you need.
Deep cycle batteries are rated according to the number of amp-hours they can deliver at different rates of discharge; typically over 8 hours, 20 hours and 100 hours. Use the 20 hour rate cuz it's closest to 24 hours.
So, let's say you work out that you need 800 amp hours of battery power, and your 12 volt deep cycle battery delivers 200 amp-hours when discharged over 20 hours.
Then you'd need FOUR 12 volt deep cycle batteries wired in parallel to meet your needs.
And, besides the four batteries (probably costing about $100 each) you also have to buy an inverter to convert the 12 volts DC into 120 volts AC so your pump can use that power.
For $500 (guessing) for the battery back-up system, you could buy a gasoline powered generator to keep the sump pump running.
That is, the cost of a battery back-up system to tie you over until you buy a generator would cost you darn near as much as the generator.
You'd be better off buying the generator.