Why does my calculator show $2^{-329} = 0?$

The precise name for the feature is underflow, which means it's less than the smallest number the registers can hold. It's kind of like overflow.

It doesn't. Your calculator can't handle a number of such a small magnitude. Specifically, $2^{-329}\approx 9.14\times 10^{-100}$, and I'm guessing that your calculator can only handle numbers of magnitude between $10^{-99}$ and $10^{99}$.

Your calculator gives you $0$ for $2^{-329}$ for the same reason that it gives you $0$ when you divide by $2$ a lot: Because $2^{-329}$ is dividing by $2$ a lot. More precisely,

$$2^{-329}=\dfrac{1}{2^{329}}=\dfrac{1}{\text{huge number}}=\text{number so small your calculator doesn't know it from }0.$$

Note that $2^{-329}$ is what you get if you start with $1$ and divide by $2$ repeatedly, a total of $329$ times.