Convincing people to get vaccinated against COVID-19 is a key societal challenge in the present times. As a first step towards this goal, many prior works have relied on social media analysis to understand the specific concerns that people have towards these vaccines, such as potential side-effects, ineffectiveness, political factors, and so on. Though there are datasets that broadly classify social media posts into Anti-vax and Pro-Vax labels, there is no dataset (to our knowledge) that labels social media posts according to the specific anti-vaccine concerns mentioned in the posts. In this paper, we have curated CAVES, the first large-scale dataset containing about 10k COVID-19 anti-vaccine tweets labelled into various specific anti-vaccine concerns in a multi-label setting. This is also the first multi-label classification dataset that provides explanations for each of the labels. Additionally, the dataset also provides class-wise summaries of all the tweets. We also perform preliminary experiments on the dataset and show that this is a very challenging dataset for multi-label explainable classification and tweet summarization, as is evident by the moderate scores achieved by some state-of-the-art models. Our dataset and codes are available at: https://github.com/sohampoddar26/caves-data