I'll just supplement Prof. Kalitvianski's answer by adding that the mantra « observables
as operators » only applies to the special kind of measurement called a « quantum measurement », which always involve amplification. Other kinds of measurement, such as measuring physical constants, are not modelled by observables either, like the speed of light, the mass of the electron, the charge on an electron, etc.
The Nobel prize winner Eugene Wigner wrote very careful articles about the theory of measurement, although it is considered superseded by many today, they are still an excellent foundation for understanding the more modern theories. They have been reprinted in the collection of very readable scientific essays of his, Symmetries and Reflections.