But almost all numbers satisfy that property.
This is the key understatement. There's a lot of irrational numbers. To get a sense of how many there are, consider the task of writing down all of the real numbers between 0 and 1. There's a lot right? Infinitely many. No way you could write them all. How about all of the rational numbers between 0 and 1. Writing the numbers at a finite rate, it would take you an infinite amount of step to write down all of the rational numbers.
But note the difference in phrasing. For the rational numbers, it would take an infinite number of steps to write down all the numbers. For real numbers, I said it was impossible. I chose a different phrasing, and for good reason.
We have this concept of counting. It goes 0, 1, 2, 3, 4... and keeps going. These are called the natural numbers. The set of natural numbers is "countably infinite." If you kept adding 1 over and over, you could eventually construct every natural number. (this is basically the definition of a "countably infinite" set)
If we look at the real numbers, and pile in $\pi$ and $\sqrt 2$ and all of their irrational friends, we have more numbers than we had natural numbers. The set of all real numbers is "uncountably infinite." There's more real numbers than there are natural numbers.
Big deal right? There's more rational numbers than natural numbers too, right? Well... not exactly.
It actually turns out that the set of rational numbers is countably infinite. There is a formal way to map all of the rational numbers onto the natural numbers. It's typically done using a diagonalizing approach:
So, as strange as it may sound there are exactly the same number of natural numbers as rational numbers, but there are more real numbers than that. This has many interesting effects deeper into mathematics, because we can use mathematical induction to prove things as long as the set of values we're proving it over is no bigger than the natural numbers. Once we move into larger sets, like the set of irrational numbers, mathematical induction is no longer a valid tool in a proof.
And yes, the ability to use mathematical induction is a big deal =)