What would/could happen if there was a problem with one cell in the body? Does the body have a repair system?

There is a 'self-destruct' button in every cell called the lymphocyte. It's a packet of enzymes that is designed to kill the cell in case of a problem, so no, it happens and is normal. The problem you run into in cancer cells is that they don't have these 'self-destruct' buttons and the cells grow out of control.