Building a foundation for hybrid cloud storage involves using object storage as a key building block and software defined storage to provide compatible infrastructure across on-premises and cloud environments. This allows for workflow mobility across locations by overcoming data gravity, seamless operations, strong data security, and flexibility in infrastructure choice. Object storage is designed to handle the growing amounts of unstructured data and needs of web-scale applications, while hybrid cloud storage integrates on-premises and cloud storage through methods like using cloud storage as remote storage, replicating and mirroring data between locations, and providing consistent APIs and workload management across environments.