The object doesn't have any depth to speak of, so it completely lacks mass. Nevertheless, while it lacks volume, it still has area, so the object could be thought of as a two dimensional plane in 3D space.
It must have a depth though, there's no possible way it could exist.
If it were possible it's volume would be (depth x area) which as you're saying would be (0 x area) therefore always resulting in 0.
Or what if a 3D object were to travel along the non existant axis of the 2D object, as the 3D object could not collide with the 2D object as it has no depth and would occupy the 2D objects space, therefore the 2D object would not be able to exist at where it is.
