Allodial title is a real theory. It has even been used in a modified form in the US, for example, in NV.
http://en.wikipedia.org/wiki/Allodial_title#United_States
In NH, as in parts of AK, there are areas without property taxes. People live in these areas. However, the areas are usually owned by larger agricultural or mining interests. For example, in NH, the people living on the land tend to be either working for timber companies or leasing their home from timber companies. These areas almost never have children in government schools living at home, as that is the main expensive when it comes to property taxes.
As for NH, there is a government charter school which is an online school. Students from around the state attend it by sitting at a computer in their bedroom. You might be able to find a low property tax town in NH with few children and widespread high speed internet use. If you could convince all of the parents to either send their children to a private school, an online charter school or homeschool, you could eliminate the majority of the already low property taxes. Then, if the town currently has police (some NH towns don't), you could have the town create a couple unpaid constable positions and contract with the state police for serious crime. Actions like those might lower the average property tax bill from $1,500 a year per family to $500 a year per family.
There are other communities in NH which pay almost no property taxes but residents in those communities contract with private companies or near-by local government for services and are in private developments which charge fees for road maintenance, water, maintaining the golf course and so on.
As for no property taxes being possible. In theory, it could be possible. A large tract of land could be developed into a private community. When people buy the land, the expensive of the land might by 20 times what similar land normally costs in the area. The extra expense could be invested. The proceeds from the investments could be processed by the developer and used to pay the area property taxes. As long as the investments were successful, the home owners may never have to personally pay property taxes.