He was born in Stavanger to merchant and captain Andreas Sømme (1788–1853) and his wife Johanne Cathrine Kastrup (1789–June 1817). In 1845 he married Johanne Margrethe "Hanna" Bulletin Kielland (1823–1908), youngest daughter of Jacob and Axeliane Christine Kielland and aunt of novelist Alexander Lange Kielland and painter Kitty Lange Kielland.
Jacob Jørgen Kastrup Sømme started his career as a businessman as a clerk in Hamburg. In 1838 he returned to Stavanger. A ship-owner, Jacob Jørgen Kastrup Sømme especially concentrated on fishing and exporting atlantic herring.
He was also vice consul for the Russian Empire in 1855 and the Netherlands in 1855.
He was Knight of the Order of Saint Stanislaus. Sømme was also involved in local politics, as mayor of Stavanger in 1859, 1860 and 1869.
Having married a member of the Kielland family in 1845, he became involved in the company of his father-in-law Jacob Kielland.
