One of my dc is a boarder at a state boarding school - dh and I both boarded at trad public schools, but wouldn't have chosen that route for our dc, who were at mainstream comprehensives. Until we had to move house and area for job reasons when youngest dc was about to go into Y9, and we all really liked the boarding option attached to the local state school in our new area.
The school is about 50% boarding (used to be more) and there's a big social mix among parents of boarders: lots of local-ish farming families who live so rurally they would have a long journey to school as a day student; quite a few African families who see it as a prestige option (although you need to be an EU citizen to qualify for a state boarding place); plenty of familes with complicated work arrangements who find boarding works for them. There are some posh families who use it as a cheap alternative to public school, but tbh not that many, as it doesn't have the prestige that a 'proper' boarding school would have, for people who care about that kind of thing. And reading between the lines there do seem to be a few students who have been placed there because other options have broken down for them - I don't know if LAs are paying fees as an alternative to taking them into care, but it seems entirely possible.
It does have some of the trappings of a trad public school - my other dc take the piss when youngest talks about 'matron' and 'prep'
, but it's not stuffy at all. Overall it's an interesting and buzzy mix, and my dc has really benefited from the boarding aspect, although we wouldn't have sought it out if it hadn't happened to fit our circumstances.
Boarding fees are about £12k a year all in, as others have said the actual education part is state-funded. If you look up State boarding Schools Association you should get a list of all the UK state schools that have a boarding element. Do your homework though, as some school have just a very few boarders, whereas for others it's a much larger part of the cohort. You are supposed to demonstrate a 'boarding need', but IME this is interpreted pretty generously - most schools are not oversubscribed for the boarding element. Also note that you generally can't convert a boarding place to a day one, to avoid people using it as a sneaky route in.