There is something very rustic and traditional feeling about a woodburning waterheater. But that tends to hide the fact that they are reasonably high tech these days.
Wood heating systems have been around pretty much forever - the fire was one of the great leaps forward made by early man. In house fireplaces have long been used (inefficiently and sometimes dangerously) for heating the home and even the water supply.
The best way to get your hot water at home using wood as a fuel is to invest in a modern wood furnace. The advantage of wood is that it is cost effective.
The disadvantage of wood power is that it is not exactly what you would call convenient. Luckily many modern systems are used to supplement say a central gas heating system, or an oil one. This means now running short of water, whilst taking advantage of the cost effectiveness of wood.
The thing to bear in mind with wood is safety (with all home heating really). It will need to be installed by an expert, and all your local codes and regulations will need to be observed. Of course you have to tell your insurance company. All that said, with due care to these details, wood can be a very fine home hot water heating fuel.