Manager Roy Hodgson, who selected England's youngest 23-man squad for Euro 2016, resigned afterwards, having won just three of his 11 matches in major tournaments.
"It's about growing as players - a lot of young players will learn from this and hopefully learn from it quickly and try and bring English football back to where it belongs," Hart told British media.
"But right now we've put it at a low point and we're going to have to take that... We've worked hard, ultimately with no success so that's how this team's going to be remembered.
"It's going to be down to digging deep and going again and trying to build something new."
With England also unsuccessful in their attempt to make it out of the group stages in the 2014 World Cup, Hart felt the squad could only blame themselves for failing to meet expectations at another major international tournament.
"As a group it's down to us. All the plans are put in place," he said. "Ultimately we didn't perform. I didn't perform personally to the level that's required to push through to the quarter-finals.
"We were in a good place - we've been in a good place in a lot of tournaments I've gone to but we haven't ultimately done it so we're going to get a lot of flak for it and we deserve it."
(Reporting by Shravanth Vijayakumar in Bengaluru; Editing by Greg Stutchbury)
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