What size power station do you need to keep Wi-Fi up for 24 hours?

Short answer: 407 Wh of rated capacity — that's the ~500–600 Wh — small class. A full day of connectivity from a surprisingly small battery.

The math, appliance by appliance

ApplianceAvg drawHours usedEnergy
Wi-Fi router + modem12 W24 h288 Wh
Energy needed per day288 Wh
÷ 0.85 inverter efficiency339 Wh
× 1.2 headroom (ageing, cold, overrun)407 Wh
Recommended: ~500–600 Wh — small class. Output check: 12 W continuous, 12 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 407 Wh work?
The bare energy need is 339 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?
These loads draw 12 W continuously with no motor surges — any unit big enough on capacity is big enough on output.
Could solar keep this running indefinitely?
Roughly 96 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.