Cantor space is used quite widely in the theory of representations of topological spaces in general, and not just the real numbers.
An important realizability model, namely Kleene's function realizability (also known as Type Two Effectivity) uses Cantor space (or equivalently the Baire space
nat -> nat) as the underlying partial combinatory algebra. This realizability model, and the related computable versions of it, provide a very good environment for studying computability on topological spaces that goes much farther than computability of the real line.
In a somewhat related line of work Martin Escardó has used Cantor space to demonstrate how certain higher-order functionals can be implemented on compact spaces, see for example his Seemingly impossible functionals for a gentle introduction.