In the era of big data, the sharing of massive amounts of data is a prerequisite for fully mining the value of data. Some of these data are sensitive data involving user privacy, and special attention should be paid to the data sharing process. However, traditional data sharing methods have defects such as unclear data flow and difficulty in accountability. To solve these problems, a blockchain-based sensitive data controllable sharing solution that supports regulation is proposed. By using the dynamic accumulator technology to achieve access control of sensitive data, the data owner can flexibly grant or revoke the access rights of other participants to the data and realize the controllability of the data by the data owner. A regulator is set up to check the data request process. The regulator will issue a regulatory certificate to the data requester after the check is approved. Only the data requester who has the regulatory certificate and is authorized by the data owner can get the data. To protect the privacy of the data requester, unrelated third parties cannot obtain the identity information of the data requester by using strong designated verifier signature (SDVS). Blockchain technology is used to record data requests and responses. The record can only be read by the regulator, which realizes the regulation of the whole data sharing process. The security analysis proves that the scheme satisfies the privacy of the data requester, the controllability of the data owner, and accountability. The simulation experiment proves the feasibility of the scheme.