Sleep monitoring plays a crucial role in maintaining good health, with sleep staging serving as an essential metric in the monitoring process. Traditional methods, utilizing medical sensors like EEG and ECG, can be effective but often present challenges such as unnatural user experience, complex deployment, and high costs. Ballistocardiography~(BCG), a type of piezoelectric sensor signal, offers a non-invasive, user-friendly, and easily deployable alternative for long-term home monitoring. However, reliable BCG-based sleep staging is challenging due to the limited sleep monitoring data available for BCG. A restricted training dataset prevents the model from generalization across populations. Additionally, transferring to BCG faces difficulty ensuring model robustness when migrating from other data sources. To address these issues, we introduce SleepNetZero, a zero-shot learning based approach for sleep staging. To tackle the generalization challenge, we propose a series of BCG feature extraction methods that align BCG components with corresponding respiratory, cardiac, and movement channels in PSG. This allows models to be trained on large-scale PSG datasets that are diverse in population. For the migration challenge, we employ data augmentation techniques, significantly enhancing generalizability. We conducted extensive training and testing on large datasets~(12393 records from 9637 different subjects), achieving an accuracy of 0.803 and a Cohen's Kappa of 0.718. ZeroSleepNet was also deployed in real prototype~(monitoring pads) and tested in actual hospital settings~(265 users), demonstrating an accuracy of 0.697 and a Cohen's Kappa of 0.589. To the best of our knowledge, this work represents the first known reliable BCG-based sleep staging effort and marks a significant step towards in-home health monitoring.