Depth imaging is a foundational building block for broad applications, such as autonomous driving and virtual/augmented reality. Traditionally, depth cameras have relied on time-of-flight sensors or multi-lens systems to achieve physical depth measurements. However, these systems often face a trade-off between a bulky form factor and imprecise approximations, limiting their suitability for spatially constrained scenarios. Inspired by the emerging advancements of nano-optics, we present Nano-3D, a metasurface-based neural depth imaging solution with an ultra-compact footprint. Nano-3D integrates our custom-fabricated 700 nm thick TiO2 metasurface with a multi-module deep neural network to extract precise metric depth information from monocular metasurface-polarized imagery. We demonstrate the effectiveness of Nano-3D with both simulated and physical experiments. We hope the exhibited success paves the way for the community to bridge future graphics systems with emerging nanomaterial technologies through novel computational approaches.