Is it possible to have motion in the absence of a net force?

It is impossible ...

Explanation:

Under no circumstances this is possible because of the Newton's 1st law of motion that states that a body shall continue its motion forever except the net fore is zero. When two opposing forces contract each other, they cancel out the action of each other and ultimately the net force become zero. If one body has a slightly more or less force then the net force will not be zero and the net magnitude of force will have a direction of motion.