Baer was born in Bavaria in 1911; originally a confectioner, he became a guard in Dachau concentration camp after becoming unemployed in 1930. In 1939, he joined the SS-Totenkopfdivision, and was appointed adjutant of Neuengamme concentration camp in 1942. Another year later, in 1943, Baer became adjutant of SS-Obergruppenführer Oswald Pohl, then chief of the Wirtschaftsverwaltungshauptamt; he took over department D I in November the same year before succeeding Arthur Liebehenschel as commander of Auschwitz from May 11 1944 until the final dissolution of the camp in early 1945.
At the end of the war, Baer fled and lived near Hamburg as Karl Neumann; he was found and arrested in December 1960, and died in detention in 1963. He was a member of N.S.D.A.P. (no. 454991) and the SS (no. 44225).