I've come across the same issue (that it's difficult to explain math to non-mathematicians) many times, but inspired by this thread I decided to think more precisely about possible causes and solutions to this problem. I was further inspired because my dad happens to be in town, and when I tried to explain set theory to him, one response I got from him was essentially the same one you got, "so you're trying to describe ultimate reality."

First, I think the following important features about mathematics in general (i.e. not just set theory specifically) are unknown to most people:

1. Math is vast - People somehow know that most academic fields are vast, but they don't know this about math. For instance, someone who has only studied up to classical mechanics has still heard about general relativity, classical mechanics, fluid dynamics, electricity and magnetism, etc. On the other hand, a lot of people honestly think there's nothing more to math than matrices and calculus.
2. Math is about cool ideas - It's about coming up with cool ideas, exploring them, figuring out facts about them, and proving this facts using both creativity and logic. It's not about crunching out numbers using complicated formulas.
3. Math is not about the real world - This is an overgeneralization to the point that it's false, but it might be closer to the truth than what your audience thinks math is about. Historically, math has absolutely been about modeling the real world and solving real world problems, but for various reasons, much of modern math is done purely for its own sake, and the content it discusses is very abstract. People will try to relate what you explain to them to something they already know, and this is an entirely natural thing to do, but it's almost surely bound to miss the point.
4. Math is new - People will often look at you quizzically when you say you study math, and ask you, "what's there left to figure out?" Although the formulas of single variable calculus that they're familiar with have been figured out for centuries, there are constantly new questions arising in math, especially since math is so vast.
5. Math is a different language - First of all, there's a lot of technical strange-sounding vocabulary. Secondly, there's a lot of technical familiar-sounding vocabulary that means something different in natural language - be careful about how you use the word "axiom" for instance.

When it comes to set theory specifically, if you get a response like, "so you're trying to describe ultimate reality," you have to spend a bunch of time convincing them that current set theory isn't really about anything that they already know or familiar with. If you manage to do this, then you'll be faced with the following question, "then what's the point?" This is where some historical motivation becomes necessary I think.

So you could explain that Cantor started the study of sets almost 150 years ago because people were starting to study the real numbers, and functions on them, in more and more sophisticated ways and talking about infinity in more and more sophisticated ways, and so this "required" a more sophisticated, rigorous framework for talking about these things. This naturally led to asking more precise questions about infinity. Thinking about the real line as set of points and not just a geometric line, and thinking about functions as objects that act on points, was quite a novel idea at the time. You can try to explain the importance of bijections to the notion of size and counting, and then state and vaguely explain that $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ have the same size, but $\mathbb{R}$ is strictly bigger. Emphasize that $\mathbb{Q}$ only seems bigger than $\mathbb{N}$ because the way it's arranged not because it actually has more things. Then introduce Cantor's Problem - is there a subset of the reals that's a bigger infinity than the naturals but a smaller infinity than the whole set of reals?

Then I'd talk about Hilbert and his famous problems, the first two of his problems being about putting math on a rigorous logical foundation. Talk about how truth and provability can be formalized, what it would mean for a formula to be true (in some model) but unprovable (from some axioms), and then talk about Godel's groundbreaking results on incompleteness. I would then come back to set theory, mentioning that a great variety of questions had come up in set theory, a good deal of them had turned out to be undecidable, including Cantor's Problem.