More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of positive 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 <a href="http://mathworld.wolfram.com/IrrationalityMeasure.html">irrationality measure</a> 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 <a href="http://math.stackexchange.com/questions/7634/you-are-standing-at-the-origin-of-an-infinite-forest-holding-an-infinite-bb-gu/">the last part of this math.SE question</a>.