The Keybase filesystem is a cloud-based encrypted, signed file system that permits its users to have public and private directories attached to their identities (e.g., privately-chosen usernames or publicly-known social network identities). Public and private encryption key sets, with the private keys known only to the individual owners (not stored by Keybase on their servers), are used to allow private files to be stored where only the owner can access them, and public files to be signed to authenticate which user posted them, with these actions taking place automatically behind the scenes once all the relevent software and keys are installed.
Client software for various platforms allows this file system to be treated as an extension of the native filesystem of whatever system you are using, and a publicly documented API allows more clients to be written by third party developers.
Currently, Keybase is offering free accounts with up to 10 gigabytes of storage, with the quota being applied to files you place online yourself (files that others place in your public inbox directory count against the uploader's quota, not yours). Paid accounts with higher capacity may be offered in the future; the intended business model is not to use advertising or sale of personal information to third parties in the future, so the protocols are purposely designed without the ability of the Keybase management to inject themselves in the middle of data transfers for such purposes.