Federated learning (FL) has emerged as a promising master/slave learning paradigm to alleviate systemic privacy risks and communication costs incurred by cloud-centric machine learning methods. However, it is very challenging to resist the single point of failure of the master aggregator and attacks from malicious participants while guaranteeing model convergence speed and accuracy. Recently, blockchain has been brought into FL systems transforming the paradigm to a decentralized manner thus further improve the system security and learning reliability. Unfortunately, the traditional consensus mechanism and architecture of blockchain systems can hardly handle the large-scale FL task due to the huge resource consumption, limited transaction throughput, and high communication complexity. To address these issues, this paper proposes a two-layer blockchaindriven FL framework, called as ChainsFL, which is composed of multiple subchain networks (subchain layer) and a direct acyclic graph (DAG)-based mainchain (mainchain layer). In ChainsFL, the subchain layer limits the scale of each shard for a small range of information exchange, and the mainchain layer allows each shard to share and validate the learning model in parallel and asynchronously to improve the efficiency of cross-shard validation. Furthermore, the FL procedure is customized to deeply integrate with blockchain technology, and the modified DAG consensus mechanism is proposed to mitigate the distortion caused by abnormal models. In order to provide a proof-ofconcept implementation and evaluation, multiple subchains base on Hyperledger Fabric are deployed as the subchain layer, and the self-developed DAG-based mainchain is deployed as the mainchain layer. The experimental results show that ChainsFL provides acceptable and sometimes better training efficiency and stronger robustness compared with the typical existing FL systems.