The blockchain-based video systems are designed to build a decentralized and flexible video ecosystem by enabling a direct interaction among users, video providers (VPs), and service providers. In blockchain-based video systems, the heterogeneous qualities and formats of the video streams usually require massive computational resources to transcode them into different versions and formats to meet distinct requirements of users. However, current blockchains cannot handle massive and heterogeneous video streaming due to limited computing capacity and long transaction times. To deal with this issue, in this paper, leveraging mobile edge computing (MEC) technology, we propose a blockchain-based MEC architecture, where small base stations (SBSs) allocate their computation as well as communication resources for providing video streaming in a distributed and secure manner. Moreover, to improve the operation efficiency, we use a series of smart contracts to enable a self-organized video transcoding and delivery service without a centralized controller. Then, users, SBSs, and VP could adjust their strategies according to the transactional information on blockchain. Moreover, we formulate the video transcoding and delivery problem as a three-stage Stackelberg game. We analyze the sub-game equilibrium in each stage and the interplays of the three-stage game. Last, we propose an iterative algorithm to solve the problem. Simulation results show that the proposed approach could obtain the good performance in terms of average time to finality (TTF), average access delay, and network cost.