This sort of thing exists and is apparently what other commenters also have in mind, but I would argue that in your example the measuring device's output is a n

This sort of thing exists and is apparently what other commenters also have in mind, but I would argue that in your example the measuring device's output is a natural number n in the range [0, 300] and that its interpretation as a fraction of a circle's circumference is just that. Clearly the existence of describable irrational numbers implies that we can describe a measurement using an irrational number. That does not make a finite measurement essentially irrational in any meaningful sense.