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What size power station do you need to run a sump pump for 24 hours?

Short answer: 2,259 Wh of rated capacity — that's the ~3–4 kWh — extra-large class. Sump pumps fail exactly when storms knock the power out. The surge draw is the sizing constraint.

The math, appliance by appliance

ApplianceAvg drawHours usedEnergy
Sump pump (1/3 HP)800 W2 h1,600 Wh
Energy needed per day1,600 Wh
÷ 0.85 inverter efficiency1,882 Wh
× 1.2 headroom (ageing, cold, overrun)2,259 Wh
Recommended: ~3–4 kWh — extra-large class. Output check: 800 W continuous, 1,800 W surge.

Fridges and freezers use their average cycling draw for energy and full compressor draw for the output check — that's why the numbers differ from the label.

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Quick answers

Can a smaller unit than 2,259 Wh work?
The bare energy need is 1,882 Wh after inverter losses; the extra 20% covers battery ageing, cold weather, and outages running long. A smaller unit works on a good day — this size works on a bad one.
What about output watts, not just capacity?
The station must sustain 800 W continuous and survive a 1,800 W start-up surge. Nearly every unit in the ~3–4 kWh clears both comfortably.
Could solar keep this running indefinitely?
Roughly 533 W of panels replaces a day's usage with ~4 hours of decent sun at real-world (75%) efficiency. Half that solar doubles your runtime instead of sustaining it.