An object experiences a net zero external unbalanced force. Is it possible for the object to be travelling with a non-zero velocity? If yes, state the conditions that must be placed on.

Yes, it is possible. According to Newton's first law of motion, an object remains in a state of rest or of uniform motion in a straight line unless compelled to change that state by an applied force.
A net zero external unbalanced force means that the object has balanced force. Hence, if it had an initial velocity then the object will continue travelling with a non-zero velocity.
So, the conditions for an object to be traveling with a non-zero velocity while experiencing a net zero external unbalanced force are:

1. The object must have an initial velocity.
2. The object must have experienced a force in the past that set it in motion, and the net force acting on it is now balanced, resulting in a constant velocity.