How is it possible to have a model of a set theory?

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I am trying to understand the basics of model theory. Before getting too deeply into it, I would like to know how it is even possible to construct a model, i.e. a structure that satisfies axioms of a given set theory, e.g. ZFC. If I understand correctly, you will need a set theory to construct such a structure. Can a set theory be used to construct a model of itself? Or do you need some kind of meta-set theory to do that?