That's fair -- intuitions are pretty personal things. It is "obviously" false to me, and Bertrand Russell fairly well articulated the reason why by calling AC the "multiplicative axiom".
If you have sets A, B with 3 and 5 elements, respectively, the Cartesian product has 15 elements.
So, if we have an infinite series of sets, each with at least one element, I'd expect there to be something in the infinite Cartesian project just by multiplication -- the number of elements should (and I say "should" here to clearly denote my intuition) be in some sense the product of each of the individual numbers of elements. (I avoided the word "cardinality" here explicitly because I'm talking intuitively).
The idea that there's a collection of non-empty sets whose Cartesian product is empty is therefore akin to saying that you can multiply numbers bigger than zero and end up with a zero result.