Chain of custody is needed to document the sequence of custody of sensitive big data. In this paper, we design a blockchain big-data sharing system (BBS) based on Hyperledger Fabric. We denote the data stored outside of a ledger for sharing as "off-state" and "big data" (referring to extremely large data) is in this category. In our off-state sharing protocol, a sender registers a file with BBS for sharing. To acquire the file, an authenticated and authorized receiver has to use transactions and interacts with BBS in four phases, including the file transfer request, encrypted file transfer, key retrieval, and file decryption. The corresponding transactions are recorded in the ledger and serve as chain of custody to document the trail of the data. Compared with related work, BBS can perform the four phases autonomously. It utilizes the permissioned blockchain, i.e. Hyperledger Fabric, for access control and can defeat dishonest receivers. We design and implement a prototype of BBS for big file sharing. Extensive experiments were performed to validate its feasibility and performance.