Edit: (Several people have said the same thing so maybe I can add something else)

I think another point of confusion might be the difference between "arbitrarily long" and "infinitely long". The number pi is infinitely long, a single integer can be arbitrarily long.

What's the difference? Pi has infinite digits. It is an irrational number. We can never "see" all the digits of pi, because no matter how far out we go there are always more digits.

With an integer, this isn't the case. There are infinite integers, but this property is defined on the set, not on any single element in the set of integers. Every integer (a single element in the set of integers) is finite. All of them, no matter what. They all have an end.

That said, what "arbitrarily long" means is that for any length you choose, there are integers that long (or longer). Say we choose a length 5, then 10000, 10001, 10002, ... all have length 5 or greater, but every one of them has an exact length. It just happens to be greater than 5. You can do this for any length: 1000, 10^1000, whatever. Anything. Choose any number as big as you can even imagine. Try to imagine numbers bigger than anything you can even imagine. There are integers that long, and infinitely more integers longer than that. But every single one of them has an exact, finite length.

There are infinitely many integers, but every integer is finite.