City lost their final group game 1-0 to Borussia Dortmund in Germany to finish rock bottom with just three points, also missing out on a place in the Europa League.
A despondent City and England keeper said he did not know what had gone wrong with the Barclays Premier League champions' European displays.
He said: "We need to address what happened."
City knew they needed to win to have any hope of third place, and Hart said the intensity had been there, but only until Julian Schieber's 57th-minute goal.
"I think it was there, but then they score and it disappeared," he said.
Hart added that City had let themselves down by finishing last in the group, no matter how tough a draw that had paired them with Real Madrid and Ajax as well as Dortmund.
"We've not qualified, we've not even got in the Europa League," added Hart.
"We wanted a win and we've let ourselves down. We wanted to do it and if that meant Europa League then great. We've only got ourselves to blame."
City boss Roberto Mancini said: "We lost our qualification tonight. We had the chance to score, but if in these moments you don't score it's difficult to win."
The Italian added: "This can help us for the Premier League. But we don't play in the Europa League and we wanted to try and play in the Europa League."