In most introductory quantum mechanics classes, we are introduced to the Dirac notation, concept of the 'state' of the system being represented as an abstract vector in the Hilbert space associated with it, and we are told that measurements of physical quantities involve the action of a Hermitian operator associated with the respective quantity on the wavefunction. Then, we are told that the result of the measurement is one of the eigenvalues of the operator, etc.
However, these measurements are supposed to be of physical observable quantities. Thus, whenever a measurement is done in classical physics, it must give rise to a physical quantity, with dimensions (by dimensions, I mean length, Energy, etc.).
Where exactly does the dimension of the quantity being measured come into the picture while discussing QM? For example, if I say that $|x_0\rangle$ is an eigenstate of the position operator $X$, then the action of the position operator in the ket is written as follows: \begin{equation} X|x_0\rangle=x_0|x_0\rangle \end{equation} Here, what is the quantity $x_0$? Is it just a pure number? Or does $x_0$ have units of length?
If $x_0$ is just a pure number, then where does the length dimension come into picture? If the value $x_0$ is has the dimensions of length, then can the operator $X$ operate on a wavevector multiplied by a quantity with physical dimensions?
Also, are operators of the form $L^2+L_z$, or $P+X$ (which refer to physical quantities with different dimensions; there are no constants being multipled with them) valid operators, and what is the justification (for them either existing or not existing)?