It would work just like running your house on a car battery, for about five minutes. My dad has 24 industrial lead acid 2V 450Ah cells which amount to 1m³ and 4.5kWh of usable storage since if you drain more than 20% you will kill the batteries too early. The 1140Wp PV array charges about 6kWh on a sunny summer day. We use about 5kWh daily in our apartment - that's no AC or electrical heating, just the basic appliances and a gas heater. He probably uses two to three times that amount, again no AC or electrical heating. Also lead acid batteries are rated for like 10y, PV retains 80% of its capacity for 25y, so you need to change 2.5 batteries during its lifetime. Batteries alone amount close to 50% of the TCO.

So good luck running the grid on batteries for more than 15 minutes. We'd probably need to cover Alaska with lithium batteries.

At least 40% base load is what is needed and one only gets that from nuclear and geothermal if we discard fossil fuels. Maybe solar thermal with molten salt storage too, dunno.