The distance to infeasibility of a conic linear system \(Ax=b\), \(x\in C\) \((C\) is a closed convex cone) is considered. The problem is studied how to describe the set of infeasible distances that are near a given feasible distancce. It is shown for a rank one perturbation that the smallest essentially infeasible perturbation can be characterized as the solution of a certain convex optimization problem. Perturbations with restricted structure are studied. The results are extended to more general systems \(Ax-b\in C\) \((C_x\) and \(C_y\) are closed convex cones).