The diffract-before-destroy method, using 50-to 100-fs x-ray pulses from a free-electron laser, was designed to determine the three-dimensional structure of biological macromolecules in close to their natural state. Here we explore the possibility of using short electron pulses for the same purpose and the related question of whether radiation damage can be outrun with electrons. Major problems include Coulomb repulsion within the incident beam and the need for high lateral coherence, difficulties that are discussed in terms of existing and future electron sources. Using longer pulses of electrons appears to make the attainment of near-atomic resolution more feasible, at least for nanocrystalline particles, whereas obtaining this information from single-molecule particles in an aqueous environment seems a more distant goal. We also consider the possibility of serial crystallography using a liquid jet injector with a continuous electron beam in a transmission electron microscope (TEM).