TITLE: Are there counter intuitive interpretations of ZF or ZFC?
QUESTION [9 upvotes]: Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" (apart from formally modelling the axioms of ZFC)?

REPLY [3 votes]: This is very late to the party, but I think it's worth noting that there are in fact specific obstacles to finding "natural" nonstandard models of set theories, coming from computability theory. To start with, we have:

There is no computable model of $\mathsf{ZF}$.

(The argument here works for other set theories like $\mathsf{NFU}$ as well, it's quite general.)
That is, there is no computable binary relation $R$ on $\mathbb{N}$ such that $(\mathbb{N},R)\models\mathsf{ZF}$. And things get even worse if we demand "nice" models: there is no hyperarithmetic model of $\mathsf{ZF}$ with no "false natural numbers" (= $\omega$-model), and the situation is even nastier for fully well-founded models (well-founded models are guaranteed to be $\omega$-models, but not conversely).
If we drop the axiom of infinity, things get better per Olivier Begassat's answer above. However, they don't get much better: Tennenbaum's theorem winds up implying that there are no other computable models of $\mathsf{ZFC}$-$\mathsf{Inf}$. (Interestingly, $\mathsf{ZF}$-$\mathsf{Inf}$ is a more complicated beast - see here.)
So while the downward Lowenheim-Skolem theorem tells us that if $\mathsf{ZFC}$ is consistent then it has a very "unintended" model (namely a countable one), we aren't going to find any particularly simple models.