Can Turing Machines decide on computability?
A real number cannot be input into a Turing machine, since it is an infinite object. There are various models for providing a Turing machine with oracle access to real numbers, but these are by definition computable.

You could imagine a Turing machine given a ZFC definition of a real number, and in that case it's undecidable. For example, given a Turing machine $T$, you could define a number $x$ as follows: if $T$ doesn't halt in $n$ steps then the $n$th bit of $x$ is zero. Otherwise, it is the $n$th bit of Chaitin's constant. So $x$ is computable if and only if $T$ never halts.