Born in the Danish town of Glostrup, goalkeeper Martin Hansen joined Liverpool from Brøndby in the summer of 2006 when he was 16-years-old, presumably hoping to have better fortune in England than his accident-prone goalkeeping compatriot Michael Stensgaard did in the 1990s. Hansen was at Liverpool's Kirkby Academy for his first year in England and was part of the 2007 team that reached the FA Youth Cup final against Manchester United and retained the trophy that had been won against Manchester City twelve months earlier. Hansen broke a finger during the warm-up that preceded the home leg of the final against United at Anfield. Although in some discomfort, he played in the match but was unable to participate in the second leg at Old Trafford a week later, which Liverpool’s youngsters won after a penalty shoot-out.

After graduating from the Academy, Hansen moved up to Melwood prior to the 2007/08 season, having just celebrated his seventeenth birthday. Earlier in the same year, Liverpool had signed another promising young goalkeeper the Australian Dean Bouzanis, described by both Liverpool’s manager and the club’s goalkeeping coach as "the best goalkeeper in the world of his age." So Hansen knew from the day he arrived at Melwood that he faced an unenviable task trying to dislodge some of the other ‘keepers who were ahead of him in the pecking-order. As things turned out, patience was the key to Hansen’s future. Reina was as consistent as ever in the first team but Charles Itandje disappeared on loan to Kavala in Greece, Diego Cavalieri was sold to Cesena in Italy, Bouzanis was loaned out to Accrington Stanley in 2009/10 and in the same season the Hungarian Peter Gulacsi was loaned to Tranmere Rovers, a move that was repeated early in the 2010/11 season. In the meantime Liverpool had bought Brad Jones from Middlesbrough. Hansen was sufficiently well thought of to be given a first-team squad number (41) for the first time in the 2010/11 season and was named on the bench when Liverpool faced Northampton Town in the League Cup at Anfield towards the end of September.

Hansen was named in six matchday squads under Roy Hodgson, four times in the Premier League, once in the League Cup and once in the Europa League. However, after Dalglish replaced Hodgson in January 2011, Hansen moved out of the first-team picture. He joined Bradford City on an emergency 28-day loan in July 2011 and returned to Viborg FF in his homeland on the last day of the January 2012 transfer window without fulfilling his dream of playing in a competitive match for Liverpool's first team.

Hansen played in thirty-two league matches for Viborg F.F. during the 2012/13 season. His club was promoted to the Danish Superliga as First Division champions. The goalkeeper moved on to another Danish club, FC Nordsjælland, in May 2013 and made 13 Superliga appearances for them in the 2013/14 season. His new club finished the season in 6th place but the club he had helped to promotion in 2012/13, Viborg F.F., finished bottom and were relegated back to the Danish first division. Hansen then moved to Dutch club ADO Den Haag and in August 2015 scored an equalising goal in a league game against PSV Eindhoven after he had gone up for a corner. Hansen was a regular in the Eredivisie for two years and in June 2016 joined German Bundesliga side Ingolstadt.

After being relegated from the Bundesliga in 2017, Hansen began the following season in the second tier but was then loaned to Heerenveen in the Dutch Eredivisie. In the summer of 2018 he agreed a two year deal with FC Basel. However after failing to establish himself as the Swiss side's number one this contract was terminated by mutual consent in June 2019. He had brief spells with Stromgodset in Norway and Hannover 96 in Germany, before being out of the game for two years. On 1 August 2022 he joined Danish side Odense BK. In his first season there he was the number one keeper but less than halfway through 2023/24, which ended in relegation, he failed to regain his place following a red card suspension. 

Odense Boldklub began the 2024/25 campaign in unfamiliar surroundings after dropping into the Danish 1st Division. They quickly set the pace in the second tier and rarely looked troubled in the race for promotion. Hansen played a pivotal role in Odense's promotion after the regular season, when the promotion phase started. He was then entrusted with the starting goalkeeper position and featured in all 10 promotion matches, showcasing his leadership and shot-stopping abilities. His performances were instrumental in OB securing the 1st Division title and earning promotion back to the Superliga.