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The measure of the set of all real numbers that are not normal is $0.$

Notice first that we can restrict attention to numbers between $0$ and $1.$ A number in that interval is normal precisely of that number plus any integer you choose is normal.

Take the probability that a number between $0$ and $1$ is in any interval $(a,b),$ where $a<b,$ to be $b-a.$

Then the digits of a number are independent random variables, i.e. the conditional probability that the $k$th digit is (for example) $6,$ given the values of some of the other digits, is $1/10$ regardless of the values of those other digits.

It follows from the <b>strong law of large numbers</b>, a standard theorem in the theory of probability, that the probability that each of the digits $0$ through $9$ occurs $1/10$ of the time is $1.$ For sequences of digits, rather than individual digits, there are complications that can be overcome in fairly routine ways. Hence the probability that a such a random number is not normal is $0.$

That does not mean there are no non-normal numbers. It's like the fact that a dart thrown at the unit square has probability zero of landing exactly on the diagonal is zero, but that doesn't mean there are not plenty of points on the diagonal.