Sea level contributions from the Greenland Ice Sheet are influenced by the rapid changes in glacial terminus positions. Also known as {\it calving front} positions this information is captured in satellite imagery, but determining the position of the actual front usually involves laborious human labor, causing a major bottleneck in processing the thousands of existing images. From Landsat satellite imagery, we face the task of generating 22,678 calving fronts across 66 Greenlandic glaciers. Automated methods face challenges that include the handling of clouds, illumination differences, sea ice mélange, and Landsat-7 Scanline Corrector Errors. To address these needs, we develop the Calving Front Machine (CALFIN), an automated method for extracting calving fronts from satellite images of marine-terminating glaciers, using neural networks. CALFIN builds upon existing neural network architectures, and specializes in the segmentation of line-like features, while simultaneously handling large amounts of noise in the source data. Novel post-processing algorithms are used to perform the feature extraction and vectorization. The results are often indistinguishable from manually-curated fronts, deviating by on average 2.25 ± 0.03 pixels (86.76 ± 1.43 meters) from the measured front. This improves on the state of the art in terms of the spatio-temporal coverage and accuracy of its outputs, and is validated through a comprehensive intercomparison with existing studies. The current implementation offers a new opportunity to explore sub-seasonal and regional trends on the extent of Greenland's margins, and supplies new constraints for simulations of the evolution of the mass balance of the Greenland Ice Sheet and its contributions to future sea level rise.