Giødesen was the son of Navy Commander Peter Ferdinand Giødesen and Laura Susanne Georgine Storch in the Danish port town of Korsør. He studied at the Royal Danish Academy of Fine Arts (Det Kongelige Danske Kunstakademi) in Copenhagen, where he was a student of Frederik Vermehren, Carl Heinrich Bloch, and Carl F. Andersen.
Giødesen traveled throughout Europe, including Germany in 1884-86, 1888, 1890; Sweden several times from 1889, Holland, Switzerland, Italy in 1890, Belgium, the Mediterranean, North Africa, Greece, Asia Minor, and the Black Sea in 1893. He was employed as a Professor of Drawing at Søofficer School from the spring of 1886 until 1907.
He exhibited widely, notably at the Chicago World's Fair in 1893 and at the official exhibition gallery of the Royal Danish Academy of Art, Charlottenborg Exhibition Hall (Kunsthal Charlottenborg).