He grew up in Trieste, where his father Carl Isidor was the director of the Marine Biological Station. In late 1914 the Cori family moved to Prague and Carl entered the medical school of Charles University in Prague. While studying there he met Gerty Theresa Radnitz. He was drafted into the Austro-Hungarian Army and served in the ski corps, and later was transferred to the sanitary corps, for which he set up a laboratory in Trieste. At the end of the war Carl completed his studies, graduating with Gerty in 1920. Carl and Gerty married that year and worked together in clinics in Vienna. Their only child, Tom, married Anne, a daughter of the American constitutional lawyer and anti-feminist Phyllis Schlafly.