A boy carrying a box on his head is walking on a level road from one place to another on a straight road and is doing no work against gravity.

The given statement is true 

The force of gravity is perpendicular to the direction of displacement of the boy. 
Work = force × displacement in the direction of force
In this case, displacement is zero along with the force because the boy is moving horizontally and the force(mg) acting downward.
Hence, zero work is done.