Intuition why $\nabla \cdot (\nabla \times H) \neq 0$ cannot be satisfied

Question

I found very understandable proofs why $\nabla \cdot (\nabla \times H) = 0$ might be zero. But I do not derive any understanding out of it.

Imagine I have any source field. Why cannot I use Stokes Theorem on it and construct a field which would have the source field as curl?