In federated submodel learning (FSL), a machine learning model is divided into multiple submodels based on different types of data used for training. Each user involved in the training process only downloads and updates the submodel relevant to the user's local data, which significantly reduces the communication cost compared to classical federated learning (FL). However, the index of the submodel updated by the user and the values of the updates reveal information about the user's private data. In order to guarantee information-theoretic privacy in FSL, the model is stored at multiple non-colluding databases, and the user sends queries and updates to each database in such a way that no information is revealed on the updating submodel index or the values of the updates. In this work, we consider the practical scenario where the multiple non-colluding databases are allowed to have arbitrary storage constraints. The goal of this work is to develop read-write schemes and storage mechanisms for FSL that efficiently utilize the available storage in each database to store the submodel parameters in such a way that the total communication cost is minimized while guaranteeing information-theoretic privacy of the updating submodel index and the values of the updates. As the main result, we consider both heterogeneous and homogeneous storage constrained databases, and propose private read-write and storage schemes for the two cases.