It is not really about measuring things, but about reaching a definitive answer given some assumptions. Sometimes our notation of numbers get in the way of writing things shortly (instead of infinite decimal places), other times we can use a fraction and be exact on the paper we write on.
I would say calculus is about solving things exactly using infinitesimals and limits. There's also plenty of equations that can only be solved numerically.
What you're saying is in the practical real world we can never measure things exactly. That's true but that's not what I got from calculus. Irrational numbers come to mind (not calculus).
I come to almost the opposite conclusion as you. It is amazing that we can solve equations in spite of infinities.
