The former European champions, the only team to have played every season in the Bundesliga since its creation in 1963, failed to win for the fourth straight league match and are now on 34 points. Mainz, also on 34 a spot above them in 15th.
Darmstadt 98 were officially relegated on Saturday after a 1-0 loss to Bayern Munich, who have already secured the title. Ingostadt are in 17th, four points behind Hamburg.
Northern club Hamburg had their chances in the first half but also had keeper Christian Mathenia to thank for keeping a clean sheet in a nervous performance.
Hamburg, who barely had a shot on target in a lacklustre second half as Mainz held on for a point, will now face Schalke 04 and VfL Wolfsburg in their last two matches.
Mainz end the season with games against Eintracht Franfurt and Cologne.
Florian Niederlechner scored twice as promoted Freiburg eased past toothless Schalke 2-0 to climb up to fifth place on 47 points and hope for a spot in next season's Europa League, with the striker having now netted 11 times.
Schalke's own chances of European football have evaporated with the club in 10th on 41 points.
(Reporting by Karolos Grohmann; Editing by Toby Davis and Pritha Sarkar)
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