This statement can actually be made stronger. Not only are the vast majority of real numbers irrational, they are transcendental[1] -- indescribable in algebraic terms. The numbers we can actually talk about and name are the countable algebraic numbers -- like the square root of two -- and a few transcendentals we know about. Not many. You could count them on your fingers and toes.

The rest of the reals -- and it is most of that vast sea -- are completely inaccessible, unnameable, forever anonymous.

[1] http://en.wikipedia.org/wiki/Transcendental_number

The set of transcendental but computable numbers is also countable, however: there is a countable number of algorithms. Therefore, the majority of the reals are uncomputable, like Chaintin's constant (the probability that a random algorithm halts). This can in turn be made stronger by noticing that the definition of Chaitin's constant is finite, and by the same argument, definable real numbers are countable. It follows that the majority of the reals are indefinable.

Seems nicely analogous to "regular matter" vs dark matter and dark energy in the Universe.

The problem is that number like e and pi are not random irrational numbers, they are very specific irrational numbers with very specific properties.