Doctor of Natural Sciences, University Freiburg, Germany, 1950.
Barner received his doctorate in 1950 at the Albert-Ludwigs-University in Freiburg. From 1957 he was professor in Karlsruhe and from 1962 at his alma mater in Freiburg, where he retired in 1989. From 1963 to 1994 he was director of the Mathematical Research Institute of Oberwolfach and from 1968 to 1977 he was president of the German Mathematical Society.
Married Maria Barbara Wieland, March 7, 1989. Married Käte Fehrenbach (deceased 1988). Children: Christoph, Michael, Eva-Maria, Andreas, Susanne, Markus, Marijke.