We consider the federated submodel learning (FSL) problem in a distributed storage system. In the FSL framework, the full learning model at the server side is divided into multiple submodels such that each selected client needs to download only the required submodel(s) and upload the corresponding update(s) in accordance with its local training data. The server comprises multiple independent databases and the full model is stored across these databases. An eavesdropper passively observes all the storage and listens to all the communicated data, of its controlled databases, to gain knowledge about the remote client data and the submodel information. In addition, a subset of databases may fail, negatively affecting the FSL process, as FSL process may take a non-negligible amount of time for large models. To resolve these two issues together (i.e., security and database repair), we propose a novel coding mechanism coined ramp secure regenerating coding (RSRC), to store the full model in a distributed manner. Using our new RSRC method, the eavesdropper is permitted to learn a controllable amount of submodel information for the sake of reducing the communication and storage costs. Further, during the database repair process, in the construction of the replacement database, the submodels to be updated are stored in the form of their latest version from updating clients, while the remaining submodels are obtained from the previous version in other databases through routing clients. Our new RSRC-based distributed FSL approach is constructed on top of our earlier two-database FSL scheme which uses private set union (PSU). A complete one-round FSL process consists of FSL-PSU phase, FSL-write phase and additional auxiliary phases. Our proposed FSL scheme is also robust against database drop-outs, client drop-outs, client late-arrivals and an active adversary controlling databases.