Baum studied philosophy in Mainz, and medicine in Essen, Freiburg and Hamburg. He obtained his doctorate in 1991, and qualified as a professor (molecular medicine) in 1999 in Hamburg. In 2000, he joined MHH as professor for stem cell biology, and was associate professor of pediatrics at Cincinnati Children’s Hospital from 2002. In Hannover, he led the Institute of Experimental Hematology from 2006, served as Ombudsperson for Good Scientific Practice and Dean of Research. He was elected President of the MHH in 2013. In 2019, Baum moved to the University of Lübeck as first full-time Vice President for Medicine and member of the Board of Directors of UKSH. Since October 2020 he is Director at Berlin Institute of Health (BIH) and belongs to the board of directors of Charité.. He is member of the presidium of the German Association of Medical Faculties.