If the government truly did not even fund the education (not even through vouchers, tax breaks, or other things), it would be very hard to come up with an alternative to public education that would be in any way viable.
The only possible way to provide education in the absence of government funding would be for private parties to fund it. This could happen in one of two ways (or a mix of both). Education could simply be paid for by the parents of the children being educated. Alternatively, education could be paid for by philanthropic organizations. Most likely, it would have to be a combination of both. The problem with this is that it is unlikely that private donations and parent payments would be sufficient to overcome the loss of tax dollars.
If we had to provide education to everyone without any government funding, the most likely possibility would be a variety of church-sponsored schools with a few secular private schools for people who did not want to go to church school. These schools would likely be very much segregated between good schools with teachers getting high pay and parents paying high tuition and worse schools where parents paid low tuition and teachers got low pay or even volunteered.
There simply is no really good substitute for public funding if we are to have a system of universal education.