Urban areas and their vertical characteristics have a manifold and far-reaching impact on our environment. However, openly accessible information at high spatial resolution is still missing at large for complete countries or regions. In this study, we combined Sentinel-1A/B and Sentinel-2A/B time series to map building heights for entire Germany on a 10 m grid resolving built-up structures in rural and urban contexts. We utilized information from the spectral/polarization, temporal and spatial dimensions by combining band-wise temporal aggregation statistics with morphological metrics. We trained machine learning regression models with highly accurate building height information from several 3D building models. The novelty of this method lies in the very fine resolution yet large spatial extent to which it can be applied, as well as in the use of building shadows in optical imagery. Results indicate that both radar-only and optical-only models can be used to predict building height, but the synergistic combination of both data sources leads to superior results. When testing the model against independent datasets, very consistent performance was achieved (frequency-weighted RMSE of 2.9 m to 3.5 m), which suggests that the prediction of the most frequently occurring buildings was robust. The average building height varies considerably across Germany with lower buildings in Eastern and South-Eastern Germany and taller ones along the highly urbanized areas in Western Germany. We emphasize the straightforward applicability of this approach on the national scale. It mostly relies on freely available satellite imagery and open source software, which potentially permit frequent update cycles and cost-effective mapping that may be relevant for a plethora of different applications, e.g. physical analysis of structural features or mapping society's resource usage.