Assume for contradiction that there exists such a line $\ell$. Consider the halfplanes on two different sides of $\ell$, one of them has at least $\left \lceil \frac{1001}{2} \right \rceil =501$ points, and there are no segments between these points, but for each point in this halfplane there are two segments incident with that point, so there are at least $2\cdot 501=1002$ segments, which establishes the contradiction. Remarks. The problem holds for all odd numbers, not only $1001$ but for even numbers there always exists such line. Also, the condition no three points are collinear can be replaced with not all points are collinear.