Recent history has seen a tremendous growth of work exploring implicit representations of geometry and radiance, popularized through Neural Radiance Fields (NeRF). Such works are fundamentally based on a (implicit) volumetric representation of occupancy, allowing them to model diverse scene structure including translucent objects and atmospheric obscurants. But because the vast majority of real-world scenes are composed of well-defined surfaces, we introduce a surface analog of such implicit models called Neural Reflectance Surfaces (NeRS). NeRS learns a neural shape representation of a closed surface that is diffeomorphic to a sphere, guaranteeing water-tight reconstructions. Even more importantly, surface parameterizations allow NeRS to learn (neural) bidirectional surface reflectance functions (BRDFs) that factorize view-dependent appearance into environmental illumination, diffuse color (albedo), and specular "shininess." Finally, rather than illustrating our results on synthetic scenes or controlled in-the-lab capture, we assemble a novel dataset of multi-view images from online marketplaces for selling goods. Such "in-the-wild" multi-view image sets pose a number of challenges, including a small number of views with unknown/rough camera estimates. We demonstrate that surface-based neural reconstructions enable learning from such data, outperforming volumetric neural rendering-based reconstructions. We hope that NeRS serves as a first step toward building scalable, high-quality libraries of real-world shape, materials, and illumination. The project page with code and video visualizations can be found at https://jasonyzhang.com/ners.