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There's a second benefit to this approach in that you could use potable water tanks and fully sealed piping. This would provide many people with water to use in the event of an emergency. Obviously they would need to balance the needs of water with its utility as a way to store energy, but so long as you conserve both in the event of a disaster, you should be able to replenish what you've used via rainwater if the system is set up to collect rainwater and filter it (except in dry areas or during a drought of course).

How long would 1 KW last for the average household, and, if 1 KW is not enough, how many cubic meters of water would the average household have to store?

How long 1 kilowatt hour would last is a difficult question - depends on how efficient the home is and what tasks need to use that energy (for example, AC is very expensive but tends to be needed most when solar energy is available). Heating and cooling are most expensive.

Our energy usage in a not particularly well insulated home but in a subtropical Spring averaged about 9kWh/day - so 1kWh (3.6MJ) wouldn't get us far - it probably barely powers the fridge! However, you'd only need the storage overnight, and use solar energy the rest of the time. In sunny climes, at least.

Again, how much water depends on how much vertical drop, but it would likely be a lot of water (if you sat one cubic tank with 4.4m sides on top of another then emptying the top into the bottom would theoretically generate 1kWh)

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