The calculation
capacity (Wh) = daily use × days ÷ inverter efficiency ÷ depth of discharge ÷ cold derating
Divide by the system voltage for amp-hours. The layout shows how many units go in series to reach the system voltage and how many strings in parallel reach the capacity.
Standard voltages
Battery packs are built from cells in series, so the system voltage is a multiple of the cell voltage. LiFePO4 cells are 3.2 V nominal: 4, 8 and 16 cells give the familiar 12.8 V, 25.6 V and 51.2 V packs of 12, 24 and 48 V systems. Sodium-ion cells are about 3.1 V nominal: 12.4 V, 24.8 V and 49.6 V. Na-ion works over a wider window (roughly 1.5–3.95 V per cell), so check that the charge controller and inverter accept its voltage range and have a sodium-ion or custom charging profile.
Depth of discharge
LiFePO4 is usually cycled between about 10% and 100% — 90% usable — for a long life. Sodium-ion tolerates deep discharge well, even down to 0 V for storage and transport, so 95% is a fair design figure. Both last thousands of cycles at these depths.
Cold
Batteries lose capacity in the cold. LiFePO4 loses little capacity but must not be charged below 0 °C without a heated or protected pack. Sodium-ion is the cold-weather chemistry: it keeps around 90% of its capacity at −20 °C and many cells accept a (slower) charge down to about −20 °C — check the datasheet. It stores a little less energy per kilogram and its round-trip efficiency is slightly lower than LiFePO4's.
How many days?
Two to three days suit most climates when the array is sized for the darkest month; five or more make sense in foggy valleys or where access is hard. More autonomy is cheaper to buy as panels than as batteries in most places — see days of autonomy.
Size the panels that recharge the bank with the off-grid solar calculator.