Detailed human surface capture from multiple images is an essential component for many 3D production, analysis and transmission tasks. Yet producing millimetric precision 3D models in practical time, and actually verifying their 3D accuracy in a real-world capture context, remain key challenges due to the lack of specific methods and data for these goals. We propose two complementary contributions to this end. The first one is a highly scalable neural surface radiance field approach able to achieve millimetric precision by construction, while demonstrating high compute and memory efficiency. The second one is a novel dataset, MVMannequin, of clothed mannequin geometry captured with a high resolution hand-held 3D scanner paired with calibrated multi-view images, that allows to verify the millimetric accuracy claim. Although our approach can produce such highly dense and precise geometry, we show how aggressive sparsification and optimizations of the neural surface pipeline allow estimations in minutes of computation time using only a few GB of GPU memory, while allowing for real-time millisecond neural rendering. On the basis of our framework and dataset, we show that our method achieves submillimetric accuracy and completeness for 77% of the points in less than 3 minutes of training time, with 68 viewpoints.