The use of satellite remote sensing for rapid and accurate monitoring of water bodies can provide holistic and dynamic information on water resources, which is of great importance for the prevention and control of floods, droughts, and other disasters. Optical satellite remote sensing data can be affected by cloudy weather, which leads to the problem of large errors in water body identification. We use Sentinel-1 data to construct a synthetic aperture radar water body dataset and propose MAG-Net, a network model that fuses multi-level and multi-scale information. The model structure consists of a multi-scale residual block as the backbone network, which is designed to enhance the ability to capture different scale features of water bodies. The ability to capture detailed information is also enhanced by introducing a global information fusion block. Finally, the decoder is designed as a full-size hop-connected structure, which improves the accuracy of narrow water body recognition by combining feature maps at different scales. Experiments show that the extraction accuracy of the proposed model on Sentinel-1 data and the generalization ability on GaoFen-3 (GF-3) data outperform the existing water body extraction models. MAG-Net achieved impressive results on the Sentinel-1 dataset, with an accuracy of 97.09%, recall of 97.50%, and F1-score of 97.25%. These scores represent significant improvements of 1.89%, 2.94%, and 2.43%, respectively, compared with the worst-performing models. In addition, MAG-Net demonstrated a high accuracy rate of 95.63%, a recall rate of 94.46%, and an F1-score of 94.18% on the GF-3 verification set.