Edit: As the comments mention, I misunderstood how to use the diagonalization method. However, the issue I'm trying to understand is a potential problem with diagonalization and it is addressed in the answers so I will not delete the question.
Cantor's diagonalization is a way of creating a unique number given a countable list of all reals.
I can see how Cantor's method creates a unique decimal string but I'm unsure if this decimal string corresponds to a unique number. Essentially this is because $1 = 0.\overline{999}$.
Consider the list which contains all real numbers between $0$ and $1$:
$0.5000 \mathord\ldots \\ 0.4586 \mathord\ldots \\ 0.3912 \mathord\ldots \\ 0.3195 \mathord\ldots \\ 0.7719 \mathord\ldots\\ \vdots$
The start of this list produces a new number which to four decimal places is:
$0.4999 \mathord\ldots$
But $0.5$ was the first number and $0.4\overline{999} = 0.5$ so this hasn't produced a unique number.
Of course my list is very contrived, I admit that it's hard to imagine a list of the reals where numbers would align nicely to give a problem like this (since some numbers have no nines). However, I can't see a good reason why such an enumeration of numbers would be impossible.