Of course, hundreds of "small scale generation" units disconnecting from the grid was one of the reasons the power went off in August 2019 stranding all those Class 700 and 717 EMUs (see below). It's a while since I looked at it (
here's the final report) but from memory National Grid had about 1GW of reserve to call on (equivalent to the largest single generator in use at the time), but within about a second of the lightning strike which started everything they had lost some 500MW (estimated as it's not metered) of "embedded small scale generation", Hornsea wind farm also disconnected, and Little Barford power station disconnected all generators over a period of about 90 seconds for a total loss of around 1,900MW.
It's possible that without the loss of embedded generation the 1GW on immediate reserve would have been sufficient to stop the frequency dipping low enough to trigger the shutdown of the trains - their power was never disconnected but because of a misconfiguration following a software update, the change in frequency caused them to "lock out", and they could not be rebooted by their drivers.
The obvious answer is for National Grid to have had additional reserve in hand, but it's almost impossible to predict how much embedded generation will disconnect itself under fault conditions. Anyway, back to the plot...
If you want to use a battery (of any description) to help power the house in a power cut you need to have a split supply, so that when you go stand-alone (disconnected from the grid) the inverter - which is unlikely to be bigger than 10kW - doesn't try to run your electric shower, kettle and oven. It's something that is often done commercially or in hospitals, for example, where some circuits are "protected" - that is connected to batteries and/or generators - and others are not.
We live in a semi-rural area where short power cuts (of a minute or less) are fairly common. Occasionally - once every couple of years - we have a longer power cut (not counting planned outages), usually when something brings down the overhead cable which runs to the single transformer which powers the whole village. I have therefore always had certain equipment running from cheap UPS units; the sort sold for computers. These (I have three at the moment) keep a light or two on and keep the network equipment, desktops and DECT phones running. I did try one on the central heating boiler a few years ago, but the old-type pump didn't like the "modified sine wave" my cheap unit was putting out. Since then I've also bought a "pure sine wave" UPS and my central heating pumps are all inverter-driven, so I suppose it might be time to try it again. The Lead-acid batteries they use only last five or six years (when you test them, they hold up for a minute where previously they'd have done 15 or more) but are fairly easy to replace, if an annoying ongoing cost.
Car batteries as grid storage are not a bad idea so long as the control system keeps a (settable by you) minimum charge level in the car. People with solar PV are already fitting
house batteries, I think because with current tariffs it's more economic to store any excess generated during the day in your own battery for use during the night than it is to export it immediately to the grid and then "buy it back" later. My idea for fitting caravans with car batteries is better still!
To go back a fairly long time, the
Centre for Alternative Technology near Machynlleth was always intended to be off-grid, and they had a whole shed full of Lead-acid batteries storing the output from their many solar panels, wind turbines and water wheels and turbines. Some years ago they reassessed this and realised that - even with the generation mix of the grid then (it used a lot more coal) - it was actually less environmentally damaging (in terms of the manufacture and disposal of hundreds of Lead-acid batteries) to "store" their generation in the grid. As a bonus it was also cheaper. I wonder if they have looked at this again more recently with the advent of Lithium batteries and the increase in renewably-generated grid electricity?