Earlier quoted context omitted.
We already have cheap "batteries" in our homes. Run the electric water heater when the power is cheap. Water in the tank stays hot for a couple days. The only piece missing here is an internet-connected thermostat for the tank that inquires about the spot price for electricity and turns on when it's cheap. The same thing applies to your HVAC system.
You're exaggerating a bit. Water heater will stay hot for a day or so, but will rapidly cool when you start using it and cold water flows in. House will be uncomfortably cold after a few hours in the wintertime if the heat is not running, and it's not really practical to overheat it when electric rates are low/free because you can only go maybe 10 degrees before it's really too warm for comfort. You'd need a heat tan…
To convert that to US units, that means a 3000sqft house somewhere were electricity is $0.15c/kWh would cost $600/year to heat from electric baseboard heaters. If you have a heat pump you could bring that down to $200/year - and that's before you even consider solar.
I live in a 5 year old apartment, and haven't even turned on the heating this season, but it's still a comfortable 20c/68F inside.