In this article, we investigate a non-localization property of the eigenfunctions of Sturm-Liouville operators $A_a=-\partial_{xx}+a(\cdot)\operatorname{Id}$ with Dirichlet boundary conditions, where $a(\cdot)$ runs over the bounded nonnegative potential functions on the interval $(0,L)$ with $L>0$. More precisely, we address the extremal spectral problem of minimizing the $L^2$-norm of a function $e(\cdot)$ on a measurable subset $\omega$ of $(0,L)$, where $e(\cdot)$ runs over all eigenfunctions of $A_a$, at the same time with respect to all subsets $\omega$ having a prescribed measure and all $L^\infty$ potential functions $a(\cdot)$ having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.