First: there are no such things as nested atoms. Yes, different kinds of atoms have different sizes, but you will never find one atom tucked inside another one.
Now, when you really think about it, "empty space" becomes kind of a tricky concept to define, and it depends on the scale at which you look at the space. For example, there's a story about a philosophy (?) professor doing a demonstration in class, where he fills a cup with marbles, asks "Is there empty space in this cup?" (and the students answer "no"), then pours sand into the cup, asks "Is there empty space in this cup?" ("no"), then pours water in and so on. The point is that "empty space" is not a specific technical term. A particular region of space could be considered empty or full or something in between, depending on which definition you're using; and definitions differ.
With atoms, what's really going on is that the space outside the nucleus is filled with the quantum fields that represent an electron. (It's slightly tautological to say that because a field fills space by definition, but let's not worry about that.) The electron itself is very very small, essentially pointlike, so it doesn't really take up any space on its own, and thus you might argue that all the volume of the atom (outside the nucleus), minus the size of one electron, is empty. But on the other hand, there's some probability of the electron being anywhere within the atom, so by another definition, it's not really accurate to say that that space is empty if there's some chance an electron could show up in it. In particular, the space within an atom is not quite the same as what physicists consider a vacuum, because there's a significant probability that you'll find an electron in the atom whereas the probability is much less in a vacuum.
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