This paper provides a first study of utilizing energy harvesting for sustainable machine learning in distributed networks. We consider a distributed learning setup in which a machine learning model is trained over a large number of devices that can harvest energy from the ambient environment, and develop a practical learning framework with theoretical convergence guarantees. We demonstrate through numerical experiments that the proposed framework can significantly outperform energy-agnostic benchmarks. Our framework is scalable, requires only local estimation of the energy statistics, and can be applied to a wide range of distributed training settings, including machine learning in wireless networks, edge computing, and mobile internet of things.