Is it possible to be an atheist Jew? Absolutely. There are countless examples of people who are Jewish ethnically but do not believe in G-d. We put ourselves sometimes in that category... especially when our favorite sports teams continue to lose.
But is it possible to be a Jewish atheist? That seems a contradiction. For one cannot acquire Jewish ethnicity, and if one converts to Judaism, doesn't that stop them from being an atheist?
A bit confusing, we know. But not much more confusing than the supposed Jewish conversion of French existentialist philosopher Jean-Paul Sartre. Sartre, who wasn't Jewish ethnically, was an atheist to the core.
Supposedly, a month before his death, Sartre showed an interest in Judaism. But, from what we can gather, an interest is all that was; Sartre did not consider himself a Jew... he was just Jew-curious.
Absurd? Confusing? Well, that's existentialism for you.