Alex works in his own home as a homemaker and full-time caretaker of his children. officially, he is:

Officially, he is not in the labor force.
He is not considered to be unemployed because in the US system, there is a difference between being unemployed and being out of the labor force. Somebody who is not in the labor force is doing something (in Alex's case, taking care of his children and the household), whereas an unemployed person is not doing any of the sort.