Mapping buildings and roads automatically with remote sensing typically requires high-resolution imagery, which is expensive to obtain and often sparsely available. In this work we demonstrate how multiple 10 m resolution Sentinel-2 images can be used to generate 50 cm resolution building and road segmentation masks. This is done by training a `student' model with access to Sentinel-2 images to reproduce the predictions of a `teacher' model which has access to corresponding high-resolution imagery. While the predictions do not have all the fine detail of the teacher model, we find that we are able to retain much of the performance: for building segmentation we achieve 79.0\% mIoU, compared to the high-resolution teacher model accuracy of 85.5\% mIoU. We also describe two related methods that work on Sentinel-2 imagery: one for counting individual buildings which achieves $R^2 = 0.91$ against true counts and one for predicting building height with 1.5 meter mean absolute error. This work opens up new possibilities for using freely available Sentinel-2 imagery for a range of tasks that previously could only be done with high-resolution satellite imagery.