In order to have an aquaponic system you need a few basic things, a tank for the fish, which can be as big as you need, a biofilter to convert ammonia into nitrates, a hydoponics system, where the plants are grown normally above the fish, and the ability to pump water back into the fish tank. Although it is interesting to note that this system has been used on the land for many years, for example in China, where fish live within the paddy fields and fertilise the rice as it grows, and also provide an addition food source.
There is no need for chemicals as the naturally occurring bacteria keep it clean. Many plants grow well in this system such as salads, herbs, chillies, watercress and spinach. Other vegetables can be grown such as tomatoes and cucumbers but this depends on the nutrient value of the water. It is also possible to raise fish within this system that you can also eat.
The positives of this system is that it can be tried on a very small scale, they can be placed anywhere, uses far less water than traditional vegetable growing and does not use or need pesticides.