I think you might want to look at something like Solid[0]. It resembles your idea, but is more general. People host their data in a personal data store (a pod, which can be either self-hosted or by a 3rd party) and Web applications read to/write from this data store. It is more general in the sense that this data can then also be used by other applications to provide their own features (which is a hard problem to tackle, since you don't want to restrict all current and future different types of data to one interface).
E.g. When you create a new blog post this is stored in a pod of whichever data provider you chose. The fact that you wrote this blog post can then be discovered e.g. on your social media, after which people can read it in their favorite blog post reader.
I find the implications of such a platform to be the most interesting thing. It effectively creates two different markets: that of data providers, which compete to provide the best service, and of application providers, which compete to provide the best features.