The set of computable numbers is actually countable (see ref. linked above). It has to be by definition because the set of finite computer-programs is itself countable.
This is the whole point of the un-reality of "real" numbers: "all" of it (= measure 1) is uncomputable except a "tiny" measure-0 set.
It is funny you say that when Turing defined Turing machines to compute real numbers (like π for example). In its original definition, a number was computable if its Turing machine did not stop. Which makes sense since π does not have a finite decimal expansion.
Today, we usually define Turing machines to decide problems and a problem is decidable if for every input its Turing machine stops with a ``yes'' or ``no'' answer. I guess this is what makes people think what you said in the quote above. Maybe this definition is more intuitive but this conclusion from it could not be more wrong.
Think about it for a second, if the computable numbers were countable there would be no uncomputable problem (Turing actually used the classic cantor diagonal argument to prove that there were uncomputable numbers)