Elliott was born in Hagerstown, Maryland. His childhood home, the Elliot-Bester House, was listed on the National Register of Historic Places in 1975. He enlisted in the US Navy as a midshipman in April 1804 and saw action in the Mediterranean Sea during the Barbary Wars between 1805 and 1807, serving on board the USS "Essex" under Commodore James Barron. In June 1807, Elliott was on board USS "Chesapeake" when Commodore Barron was forced to allow a search of the ship by HMS "Leopard".