This paper is concerned with the feasibility of the power flow in DC power grids with constant power loads. We introduce the notion of distance to infeasibility as a voltage stability index and robustness measure for power flow feasibility. In particular, we study the p-norm distance to infeasibility in the domain of the constant power loads, and show how this distance may be expressed as a mathematical program. Necessary and sufficient matrix inequalities are presented that guarantee a minimal p-norm distance between a given vector of power demands and the boundary of infeasibility. For the cases 1-norm and ∞-norm distance we show that the condition can be formulated as (multiple) linear matrix inequalities, whereas in all other cases the matrix inequalities are strictly concave and thus non-convex. For the 2-norm distance we show that the distance to infeasibility may be computed via bilinear matrix inequalities. A numerical example for the ∞-norm distance to infeasibility for the 39 bus New England power grid is provided.