Oh but they do. When taking measurements in a scientific setting, it usually gives the most accurate and simple model to assume that your measurement is an approximation of a certain real number, and understand more accurate measurements as better estimates of this real number. Using rational numbers or other countable sets in this position usually leads to undesirable biases and or circuity in the model. When you need a "continuum", the real numbers have the best properties for the job.

Let's say you've got a light-emitting flat surface and want to describe its power output. I recommend modeling it as a set of real coordinates with an intensity value at each point and integrating to compute the total power output. By all means I invite you to come up with a better model, without using uncountable sets, for which there isn't a straightforward analogue, just as good, that uses uncountable sets.

Edit: Might as well jump straight to explaining how wave functions work without using uncountable sets, and why that's a better description of reality.