Current Earth observation benchmarks focus on measuring performance on diverse tasks and applications, typically measuring generalization in-distribution. But when models are deployed, they must generalize to myriad out-of-distribution scenarios, such as new time periods, geographies, scales, and sensors. We introduce EarthShift: the first public testbed for benchmarking robustness across multiple realistic distribution shifts encountered in remote sensing. EarthShift enables users to measure distributional robustness by comparing performance in- and out-of-distribution using datasets from paired datasets from different sources, temporal windows, geographic locations, and sensors. Our experiments on 8 geospatial foundation models (GFMs) and 11 tasks covering 5 shift types show that GFMs consistently perform 15-20% worse out-of-distribution on average regardless of model architecture, size, pre-training or fine-tuning strategy. We show that GFM robustness is similar to that of generic vision foundation models, and even fully-supervised models. This highlights a need for future research to strive for improvements in distributional robustness, not just performance, which can be benchmarked using EarthShift. We release our code and datasets to provide a testbed to guide future work to create foundation models that are robust and reliable in real-world applications. Code and data for EarthShift are available at: https://earthshift.github.io