The rapid growth of Web3.0 is transforming the Internet from a centralized structure to decentralized, which empowers users with unprecedented self-sovereignty over their own data. However, in the context of decentralized data access within Web3.0, it is imperative to cope with efficiency concerns caused by the replication of redundant data, as well as security vulnerabilities caused by data inconsistency. To address these challenges, we develop a Trustworthy Decentralized Cooperative Caching (TDC-Cache) framework for Web3.0 to ensure efficient caching and enhance system resilience against adversarial threats. This framework features a two-layer architecture, wherein the Decentralized Oracle Network (DON) layer serves as a trusted intermediary platform for decentralized caching, bridging the contents from decentralized storage and the content requests from users. In light of the complexity of Web3.0 network topologies and data flows, we propose a Deep Reinforcement Learning-Based Decentralized Caching (DRL-DC) for TDC-Cache to dynamically optimize caching strategies of distributed oracles. Furthermore, we develop a Proof of Cooperative Learning (PoCL) consensus to maintain the consistency of decentralized caching decisions within DON. Experimental results show that, compared with existing approaches, the proposed framework reduces average access latency by 20%, increases the cache hit rate by at most 18%, and improves the average success consensus rate by 10%. Overall, this paper serves as a first foray into the investigation of decentralized caching framework and strategy for Web3.0.