More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of integers such that


$$\left| \frac{p}{q} - r \right| < \frac{\epsilon}{q^2}.$$


I think that this is impossible if $r$ is a quadratic irrational. On the other hand, it's certainly possible for any number with [irrationality measure](http://mathworld.wolfram.com/IrrationalityMeasure.html) strictly greater than $2$.


What I really want to know is if the real numbers which don't have the property above have measure zero. If that's true, it would answer [the last part of this math.SE question](https://math.stackexchange.com/questions/7634/you-are-standing-at-the-origin-of-an-infinite-forest-holding-an-infinite-bb-gu/). 



What numbers can be approximated "pretty well" by rationals?
This is a well-studied question in diophantine approximation. You can look up Markov or Lagrange spectrum for a "description" of the numbers for which you cannot take $\epsilon$ arbitrarily small. For the answer to your last question, look up Khinchin's theorem (the answer is no, they have full measure).


