Jacobsen was born in Thisted in Jutland, the eldest of the five children of a prosperous merchant. He went to school in Copenhagen and was a student at the University of Copenhagen in 1868. As a boy, he showed a remarkable talent for science, in particular botany. In 1870, although he was already secretly writing poetry, Jacobsen adopted botany as a profession. He was sent by a scientific body in Copenhagen to report on the flora of the islands of Anholt and Læsø.