backup electrical power supply?

[parietal]

It is possible to buy a backup battery that will keep a heating system working?

my parents live in a rural location which suffers frequent power cuts (no idea why) lasting 5 mins - 5 hrs. Each time the power goes out, they have to rest all their internet systems and clocks on eletronic gadgets and they lose heating & hot water. the heating etc is oil-fired, but just needs the electric control box to be active in order to keep working.

So it is possible to get some kind of back-up battery that would keep the boiler control box working in the event of a power cut so the house does not get cold. And ideally something that would power the internet box too.

Any recommendations for how this can be done?

Just get a diesel generator that can kick in when the electric goes down. It should power your freezer, internet and the electric thing that people need to fire their gas or oil boilers. It could do very littl more than that. £1800 - £2,800 to sort out and cheap at the price.

Tesla battery system?

they had a diesel generator in a previous house but there is nowhere to site it here. and it is big & noisy & expensive. I'm hoping for a cheaper modern solution like batteries, but something that will plug in only to the correct applicances.

You would need a UPS to support the controls & router until the backup generator kicks in but such systems are widely available.
Really it depends on how far you want to go and how much you want to spend.

but is it possible to get a UPS that lasts 4 or 5 hours? that should be much cheaper and no need for a generator. but I can't tell from the websites how long their power supply would last.

A UPS to support the controls & router until the backup generator kicks in wouldn't need to last hours.

I don't know, but I suspect you'll struggle to find a battery-based solution which will do what you need. I wonder if generators may have come on a bit though, especially if you're selective about what it needs to power. If it's just to support the heating controls, fridge/freezer and low-energy lighting then it might be smaller - and quieter - than you think.