X-ray imaging techniques have proven invaluable to study biological systems at high resolution due to the penetration power and short wavelength of this radiation. In practice, the resolution and sensitivity of current X-ray imaging techniques are not limited by the performance of optics or image-recovery methods but by radiation damage. We propose the use of Compton (inelastic) X-ray scattering for high-resolution cellular imaging and provide a study of a scanning microscope geometry that requires a dose to achieve a given resolution that is three orders of magnitude lower than for coherent (elastic) scattering. We find that the dose per imaging signal is minimized at a photon energy of 64 keV. This corresponds to a short enough wavelength (0.02 nm) to provide nanometer transverse resolution and micrometer depth of field for tomographic imaging of whole cells. The microscope could be implemented at future high-energy and high-brightness synchrotron-radiation facilities to provide images of unsectioned and unlabeled cells in their native conditions at enough detail to bridge the techniques of super-resolution optical microscopy and cryo-electron microscopy. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (110.7440) X-ray imaging; (340.7460) X-ray microscopy; (170.3880) Medical and biological imaging.