Querying both structured and unstructured data has become a new paradigm in data analytics and recommendation. With unstructured data, such as text and videos, are converted to high-dimensional vectors and queried with approximate nearest neighbor search (ANNS). State-of-the-art database systems implement vector search as a plugin in the relational query engine, which tries to utilize the ANN index to enhance performance. After investigating a broad range of hybrid queries, we find that such designs may miss potential optimization opportunities and achieve suboptimal performance for certain queries. In this paper, we propose CHASE, a query engine that is natively designed to support efficient hybrid queries on structured and unstructured data. CHASE performs specific designs and optimizations on multiple stages in query processing. First, semantic analysis is performed to categorize queries and optimize query plans dynamically. Second, new physical operators are implemented to avoid redundant computations, which is the case with existing operators. Third, compilation-based techniques are adopted for efficient machine code generation. Extensive evaluations using real-world datasets demonstrate that CHASE achieves substantial performance improvements, with speedups ranging from 13% to an extraordinary 7500 times compared to existing systems. These results highlight CHASE's potential as a robust solution for executing hybrid queries.