Well. That depends on whom you might ask this.

Set theory might be inconsistent. In particular $\sf ZFC$ and its extension by large cardinal axioms. It's a nontrivial thing, to feel safe with these theories, and it takes a lot of practice and time until you understand that $\sf ZFC$ is self-evident to some extent, and [some] large cardinal axioms are somewhat self-evident as well. But of course, we might be tricked. Crooks have been known to seem honest, until you're left with a big heap of nothing in your hands. That is why Nigerian royalty is going to have a very hard time emailing people around the world. If that happens, we need to ask ourselves where the problem lies. Is it in one of the axioms, or maybe specifically in the existence of infinite sets? Will a slightly weaker set theory (e.g. $\sf ZC$) work better, or maybe we have to resort to arithmetic theories to fix things?

Our grasp of things is usually largely inconsistent,1 but we do like to think in "types". So working inside different categories when needed, or working with some type theory or another. A lot, and I mean a lot, of people will twitch when you ask them what are the elements of $\pi$. Or whether or not $\frac13$ is a subset of $e$. Of course, those who have a firm understanding of this know that this is a question of implementation, and this is like asking whether or not the machine code of one implementation of an algorithm is the same or different of another. But people don't think about it this way, although in a way they kinda do. Instead, people focus on their math, and they just remember (or more accurately: they don't) that you can formalize this in terms of set inside set theory.