Fjeldsted was born in Sundbyvester in Copenhagen, Denmark. In 1851, he and his wife were taught about Mormonism from LDS Church missionary and apostleErastus Snow. They became members of the LDS Church in 1852. The following year, he was ordained an elder in the church by Peter O. Hansen and he came to preside over districts of the church in Amager and Ålborg, Denmark and Christiania (now Oslo), Norway.
In 1858, Fjeldsted and his family emigrated to Utah Territory to join the gathering of Latter Day Saints in Salt Lake City. From 1867 to 1870, Fjeldsted served a church mission to Denmark and Norway. In 1884, he became a member of the church's First Seven Presidents of the Seventy[broken anchor]. As a general authority of the church, Fjeldsted served two more missions to Scandinavia, including two terms as the president of the Scandinavian Mission of the church from 1888 to 1890 and from 1904 to 1905. Fjeldsted also served missions in northern Utah and in Chicago, Illinois among Scandinavian emigrants.
Shortly after returning to Utah from Scandinavia, Fjeldsted died in Salt Lake City. He was a practitioner of plural marriage and had four wives and fifteen children.