5 gallons is too small for a Betta, or any fish for that matter.
Get yourself some little invertebrates such as Ghost Shrimp or Zebra Nerites. Can be just as entertaining as fish (far more so, in fact, than a Betta in a tank that small), and will be content living in a small tank.
Tiny aquatic amphibians, and their tadpoles, may also do well.
Remember, though, you do need a filter and heater for most animals and plants. Water changes could replace a filter, however, if they were performed very frequently. Also temperate species do not require a heater.
-Lynden