If you follow that line of thinking, you don't need any computation to represent real numbers; you can just make a pair of lines denote their distance (in some units). (If you don't want to be restricted by the paper size, you can map $(0,1)$ to $\mathbb R$.)
The conventional theory of computability is developed with the practical restrictions on alphabets in mind. Certainly nothing keeps you from developing a theory of computability in which the cardinality of the alphabet is that of $\mathbb R$. You could then ask for instance which real-valued functions are specifiable in that alphabet. Or you could allow entire graphs of real-valued functions to be used in the alphabet, and then you could ask which more complicated objects, such as functionals of such functions, are specifiable.