If $\rho$ is the density of the material of a uniform rod and $\sigma$  is the breaking stress, and the length of the rod is, such that the rod is just about to break due to its own weight when suspended vertically from a fixed support, then:

The stress is max at the uppermost point and is equal to the weight of the rod divided by the area. At the highest point stress = breaking stress = $\sigma$ = weight of rod/area of cross section of the rod  $=\frac{AL\rho g}{A}=L\rho g$.
$\therefore L=\frac{\sigma }{\rho g}$ .
The stress decreases linearly to zero at the lowest end.