Fettich finished high school in Szombathely and Budapest. In, 1921, he graduated from the Eötvös Loránd University in Budapest in a doctorate in arts, with a thesis about votive tablets in the Roman province of Pannonia. From 1921 to 1923, he was a student of the flute department of the Franz Liszt Academy of Music. He, in 1926, worked in the Hungarian National Museum for Numismatic and antiquities collection of the charge of the Migration Period. Having learned Russian, he was posted as one of the only archaeologists of the Soviet Union from 1929 to 1935. He was the founding editor of Folia Archaeology in 1939. In 1941, he was appointed as the director of the Hungarian National Museum. In 1941, he became a goldsmith. He made many history-themed reliefs. After his retirement in 1945, he became a physical worker, but as still worked as a goldsmith. In 1956, he was the author of several scientific papers. From 1957 to his death, he was a member of the Creative Union of Goldsmith Artists. In 1957, he participated in Expo 58 in Brussels. From 1959 to 1962, he was a member of the Hungarian Academy of Sciences as a contractual employee of Archaeological Research Group. He was awarded the Hungarian Archaeology and History of Art Medal of the Science Society and was a member of the Helsinki Finno-Ugric Company. Fettich attended several archaeological excavations.