The club is a result of a merger in 1993 between Eiðis Bóltfelag, who were founded in 23 February 1913, and Streymur, founded in 1976. Prior to the merger, both EB Eiði and Streymur usually played their football in the second or third division. After the merger in 1993 the team still resided in the lower divisions, but this soon changed. With a newly built football ground at Mølin, Eiði, and a healthy injection of young talent, EB/Streymur chased promotion to the top flight. Among these players in were: Hans Pauli Samuelsen, Bárður Olsen (Haldórsvík), Brian Olsen (Tjørnuvík), Marni Djurhuus, Gert Hansen, Arnbjørn Hansen (Kollafjørður). With the help of veteran attacker Jón Sigurd Gleðisheygg, the team won promotion in the 1999/00 season, finishing in place second, behind B36.
EB/Streymur had since gained a foothold in the top division (Formuladeildin), as a title contenders. In 2006, following a mid-table finish the previous season, the club made a strong start to the season under Polish coach Piotr Krakowski, and led the league at the mid-summer break, having recorded a club record 8–2 win against B68 Toftir. With four matches left to play their lead at the top of the table was seven points. A dip in form followed, but the club went into the final match requiring a draw against KÍ to win the title. However, a late KÍ goal, coupled with HB taking the lead in their match with two minutes remaining meant EB/Streymur were denied the title. In 2007 the club again finished second, and played in European competition for the first time, losing 2–1 on aggregate to Finnish club MyPa in the UEFA Cup first qualifying round.
Prior to the 2008 season, EB/Streymur had never won the Faroese championship neither before nor after the merger, however on 15 August 2007 and again on 14 June 2008, EB/Streymur won a trohpy for the first time by winning the Faroe Islands Cup and its second by winning the Cup again as holders the year after. On 21 October 2008, after finishing as runners-up in the league two years in a row previously, the club finally won its first championship, and thus the double in the 2008 season.
In July 2008 the club was drawn against Manchester City of England's Premier League in the first qualifying round of the UEFA Cup 2008-09. Director Rólant Højsted announced that the home leg of the tie would be held at one of the larger grounds in the Faroe Islands, either Toftir or Tórshavn, as the club's við Margáir ground has a capacity of only 1,000. The tie, which was played in Tórshavn, ended with a defeat of 0 – 2 to Manchester City with goals coming in the first half from Martin Petrov & Dietmar Hamann. The away leg was played at the Oakwell Stadium of Championship League side Barnsley Football Club rather than Manchester City's, City Of Manchester Stadium due to the pitch being re-surfaced. The Pitch was re-surfaced as a Bon Jovi Concert was a couple of weeks before the match.
EB/Streymur and FC Suðuroy in the Faroese Cup match on 25 April 2010, which EB/Streymur won 4–1. Four months later EB/Streymur won the Cup (Løgmanssteypið). The EB/Streymur players in this photo are Gert Aage Hansen and Marni Djurhuus.