Libertarianism assumes the presence of many cultural conditions that cannot exist without pervasive free education. A Libertarian society would therefore lack the necessary pre-conditions of a Libertarian society.
Let's accept for the sake of argument that we need people to be educated. That no more requires "pervasive free education" provided by the state, than we need "pervasive free food" to prevent us starving, or "pervasive free petrol" to move around. If something is that important, people will pay for it.
Ah, but private education is expensive - thousands of pounds a year. Yes and no. Because state education is free, private education is mostly aimed at those who aren't too worried about price. But there are a lot of exceptions. Tens of thousands of children in the UK get private tutoring in addition to their schooling, and I'm aware of a number of cases where one hour a week of tutoring (at a cost of £25 or so) is enough to move a pupil from bottom of class to top of class in one or two subjects.
James Bartholemew claims that in the mid 19th century, before state education was introduced in Britain, "over 95%" of children got 5-7 years of education, mostly at charitable free or low-cost schools. I'd like to see his source for that, but I'll probably have to buy his book, which I've never got round to doing.
(5-7 years isn't a lot by modern standards, but it's as much as was needed at that time. We're a lot richer now, and could pay for more education if it were efficient and beneficial).
Modern education, as I've mentioned before, is expensive because it's based around the idea of looking after the children, all day, every day. That's for good reason, but not any reason to do with education. If you were trying to make most efficient use of teaching resources, rather than just allowing parents to go to work, six or seven hours a week would be sufficient for children to keep up the same standard as they currently do at school.
Of course, if we were to move to a cheap, efficient market-based education system, we would be left with the problem of what our children were to do all day. I would favour them working, at least from the age of 12 or so, but there are in fact many possibilities. We have a nasty coordination problem at the moment. Because everyone who cares about their children's safety sends them to school, if they are not there, they are on their own, and at some risk. As long as children are together, doing something with some kind of adults involved, they are at least as safe as they are at today's state schools.