Well, I know people who do it, but it undoubtedly takes a lot of work and organisation and probably some luck as well! I never put more than a day and a half a week into my business, which I did for a few years when I had just one child. That worked well. I gave it up when pregnant with my second child, on the assumption that it would be more juggling than I wanted to do - I am not a good multitasker!! (As it turned out, my second child was about a hundred times easier than my first. I had known that I might have carried on with the business!) I think you'll have to try it and see.
However, I'll mention a couple of things which might not have crossed your mind.
Educating your children doesn't necessarily take a large amount of time. Home education is so much more efficient than large-group instruction with a standardised curriculum that even if you choose to do formal academic work with your kids, an hour or two a day is plenty. The challenge for working parents who home educate is childcare rather than education. A childminder is a popular option, which is what I used. If my first child had been the sort of kid who would play happily for a few hours at a time (like my second child!) I could have just got on with my work while keeping a bit of an eye on her. But she needed somebody to play with or interact with pretty much all the time, so I sent her to a CM and that worked out great for all of us. We used home educating CMs so she'd have older kids around in the daytime, but you could use a non-HE childminder. If your family qualifies for the childcare element of Tax Credits, you can use that to subsidise the cost.
Nearly all children get more self-sufficient as they get older. You might not be able to look after the kids yourself while working from home when they are 4/5/6, but you could probably do it when they are preteens.
Reciprocal childcare arrangements where you look after someone else's kids one day a week and they have yours one day a week sound great, but they rely on everything coming together perfectly. It seems to me that such arrangements are better for supplementary childcare rather than as the cornerstone of your childcare. If your working hours are flexible, you may be able to assemble various extra chunks of time in which to work by childcare swaps and playdates and arranging to take one another's children to home ed groups. Obviously the fewer kids you have, the easier it is to palm them off on other families make such arrangements.
I'm sure other people will come along soon with more views on it.