He took his secondary education in Trondheim in 1920, and then moved back to Tromsø. He started a newspaper career, as subeditor of Tromsø Stiftstidende from 1921 to 1922. He was editor-in-chief in Vesteraalens Avis from 1922 to 1928 and Tromsø Stiftstidende from 1928 to 1929.
He lived as a trapper in north-eastern Greenland from 1929 to 1934.
In 1935 he was hired as secretary for Norges Svalbardog Ishavsundersøkelser, the Norwegian institution for exploration of Svalbard and the Arctic Sea, later renamed into the Norwegian Polar Institute. During World World War II, he first fled to London where he worked as a secretary for the exiled government.
From 1941 to 1944, he served with the Royal Norwegian Air Force-in-exile at Little Norway, Canada. In 1944, with the rank of Major, he was sent to Northern Norway to participate in the successful liberation from Nazi occupation.
In 1947 he returned to the Norwegian Polar Institute as secretary, and from 1948 to 1960 he was office manager.
However, he was still involved in the field as well, leading the wintering party of the Norwegian-British-Swedish Antarctic Expedition from 1949 to 1952. Giæver published several books His literary career began with Illgjæringsmann (A Misdeeder) (1921), which was translated into German in 1923.
His book Maudheim.
To år i Antarktis (1952), describing the Antarctic Expedition was translated into English, French, Dutch, Danish, Swedish, Finnish, German and Croatian. From 1955 on he renewed his literary authorship with numerous books of documentary and partly autobiographical topics, covering Arctic trapping, fishing, warfare, as well as childhood memoirs from Tromsø. Books include Ishavets glade borgere (1956).
Langt der oppe mot nord (1958), Rabagaster under polarstjernen (1959), Fra minimum barndoms elv til fjerne veidemarker (1960), Fra Little Norway til Karasjok (1964), Medical rev bak øret (1965), Dyretråkk og fugletrekk på 74° nord (1967), Lys og skygger i sjøgata (1969), Den gang jeg drog av sted (1970) and Soldøgn og mørketid (1971).
He died in November 1970 in Oslo.
He also received the King"s Medal of Merit in gold and he was a member of the Explorer"s Club in New New York