Blockchain has been applied to data sharing to ensure the integrity of data and chain of custody. Sharing big data such as large biomedical data files is a challenge to blockchain systems since the ledger is not designed to maintain big files, access control is an issue, and users may be dishonest. We call big data such as big files stored outside of a ledger that includes the blockchain and world state at a blockchain node as "off-state" and propose an off-state sharing protocol for a blockchain system to share big data between pairs of nodes. In our protocol, only encrypted files are transferred. The cryptographic key is stored in the world state in a secure way and can be accessed only by authorized parties. A receiver has to request the corresponding cryptographic key from the sender to decrypt such encrypted files. All requests are run through transactions to establish reliable chain of custody. We design and implement a prototypical blockchain off-state sharing system, BOSS, with Hyperledger Fabric. Extensive experiments were performed to validate the feasibility and performance of BOSS.