The Carlsbau in Glottertal near Freiburg, which was built between 1913 and 1914, was the outside of the Black Forest Clinic. Professor Brinkmann's house was the Hüsli local history museum in Grafenhausen-Rothaus.
Other places in the Black Forest served as a backdrop in some episodes, e.g. the historical old town of Schiltach, Lake Schluchsee, the Triberg Waterfalls and Lake Titisee.
All interior scenes were shot in the studio in Hamburg – so you won't find Professor Brinkmann's bedroom in the Hüsli local history museum (which is well worth seeing by the way).