Multilingual speech translation (ST) and machine translation (MT) in the medical domain enhances patient care by enabling efficient communication across language barriers, alleviating specialized workforce shortages, and facilitating improved diagnosis and treatment, particularly during pandemics. In this work, we present the first systematic study on medical ST, to our best knowledge, by releasing MultiMed-ST, a large-scale ST dataset for the medical domain, spanning all translation directions in five languages: Vietnamese, English, German, French, and Simplified/Traditional Chinese, together with the models. With 290,000 samples, this is the largest medical MT dataset and the largest many-to-many multilingual ST among all domains. Secondly, we present the most comprehensive ST analysis in the field's history, to our best knowledge, including: empirical baselines, bilingual-multilingual comparative study, end-to-end vs. cascaded comparative study, task-specific vs. multi-task sequence-to-sequence comparative study, code-switch analysis, and quantitative-qualitative error analysis. All code, data, and models are available online: https://github.com/leduckhai/MultiMed-ST