I think[0] we should probably be aiming for a few hours of storage, even with every affordable grid interconnect (which would help
a lot with short winter days). We use less power at night — and some of what we do use is incentivised by lower nighttime electricity costs from power plants that don’t scale down well — but I expect we will continue to use
some at night forever.
For the sake of Fermi estimation, I assume average electricity use in a developed nation is 1kW/person, and that average all-forms power use in the same is 5kW/person.
Transportation is almost certainly going to be batteries or synthesised fuels like hydrogen, and can only be backed by gravity storage if you have the kind of beamed power that would be banned by international treaty on the grounds of being too easily weaponizable[1].
If you’ll permit me to assume grid interconnects and lower nighttime demand than at present due to different pricing incentives, we might be able to need a mere 3kW-hours/person of electricity storage per night.
Using the lifetime cost estimates in the article, 3kWh/person/night is about $0.51/person/night for gravity storage and $1.10/person/night for LiIon. Neither is bank-breaking, but cheaper is better.
However, the volume required is a different question: 3kWh of batteries has a volume of 4.3 to 12 litres, while 3kWh of gravity storage is 1.1 metric ton kilometres[2].
You can make the distance required smaller by using more mass, but if I assume the average home mass is about 200 tons, you’d still have to raise the entire building 5.5 meters every day to store the same energy as a backpack of batteries. I do not expect construction on this scale to be the optimal solution in general, despite it being a very good idea in some specific cases such as hydro dams and preexisting deep shafts.
[0] Armchair opinion — I’m a software engineer not a civil engineer.
[1] Assuming they’ve read or watched any Larry Niven, the Bobbiverse, The Expanse, Babylon 5, or the news at any point in the decade following the second week of September 2001 — 'A reaction drive's efficiency as a weapon is in direct proportion to its efficiency as a drive.'
[2] http://www.wolframalpha.com/input/?i=3kWh%2F%289.8m%2Fs%2Fs%...