You need set theory to have measure theory and you need measure theory to have the analysis required to support, for example, [Fourier series][1]. Really, most of what is going on in real analysis (and hence in calculus) depends on having a predictable understanding of how infinite sums, sequences, and sets behave. 

So, elementary set theory and the ideas about infinite sets in particular are crucial for all kinds of "practical" math. 

  [1]: https://mathworld.wolfram.com/FourierSeries.html