City finished bottom of Group D, with their return of three points representing the worst ever performance from an English team in the competition's group stage.
"We have let ourselves down. We need to be winning," said Hart.
"I think Borussia and Real Madrid have done fantastic but we should be fighting until the last day to be in the competition and we weren't.
"We should have been in their position and we weren't. We've only got ourselves to blame.
"We've been disappointing in Europe but we've been fantastic in the league and consistent.
"So I don't think there are any worries about that but we need to address it to get in the Champions League again. Hopefully we can win the league then we'll be able to deal with it next year."