An outstanding challenge in single-molecule localization microscopy is the accurate and precise localization of individual point emitters in three dimensions in densely labeled samples. One established approach for three-dimensional single-molecule localization is point-spread-function (PSF) engineering, in which the PSF is engineered to vary distinctively with emitter depth using additional optical elements. However, images of dense emitters, which are desirable for improving temporal resolution, pose a challenge for algorithmic localization of engineered PSFs, due to lateral overlap of the emitter PSFs. Here we train a neural network to localize multiple emitters with densely overlapping Tetrapod PSFs over a large axial range. We then use the network to design the optimal PSF for the multi-emitter case. We demonstrate our approach experimentally with super-resolution reconstructions of mitochondria and volumetric imaging of fluorescently labeled telomeres in cells. Our approach, DeepSTORM3D, enables the study of biological processes in whole cells at timescales that are rarely explored in localization microscopy.