Cartesian product of two languages is NOT a language, hence, Cartesian product operation is NOT defined over languages, even though they are sets.

provided that the cartesian product of two languages is not an empty set.

For example, Consider, a well-defined non-empty set $L_1$ and a well-defined empty set $L_2= \phi$ defined over some non-empty input alphabet.

Then, $L_1 \times L_2 = L_1 \times \phi = \phi$

and $\phi = \{ \}$ is a language and we can make finite state machine for it.

$L_1 \times L_2 = \phi$ iff $L_1 = \phi$ or $L_2 = \phi$ or both are $\phi$.