The COVID-19 pandemic continues to bring up various topics discussed or debated on social media. In order to explore the impact of pandemics on people's lives, it is crucial to understand the public's concerns and attitudes towards pandemic-related entities (e.g., drugs, vaccines) on social media. However, models trained on existing named entity recognition (NER) or targeted sentiment analysis (TSA) datasets have limited ability to understand COVID-19-related social media texts because these datasets are not designed or annotated from a medical perspective. In this paper, we release METS-CoV, a dataset containing medical entities and targeted sentiments from COVID-19 related tweets. METS-CoV contains 10,000 tweets with 7 types of entities, including 4 medical entity types (Disease, Drug, Symptom, and Vaccine) and 3 general entity types (Person, Location, and Organization). To further investigate tweet users' attitudes toward specific entities, 4 types of entities (Person, Organization, Drug, and Vaccine) are selected and annotated with user sentiments, resulting in a targeted sentiment dataset with 9,101 entities (in 5,278 tweets). To the best of our knowledge, METS-CoV is the first dataset to collect medical entities and corresponding sentiments of COVID-19 related tweets. We benchmark the performance of classical machine learning models and state-of-the-art deep learning models on NER and TSA tasks with extensive experiments. Results show that this dataset has vast room for improvement for both NER and TSA tasks. With rich annotations and comprehensive benchmark results, we believe METS-CoV is a fundamental resource for building better medical social media understanding tools and facilitating computational social science research, especially on epidemiological topics. Our data, annotation guidelines, benchmark models, and source code are publicly available (\url{https://github.com/YLab-Open/METS-CoV}) to ensure reproducibility.