Karim Benzema's second-half goal secured Real Madrid a comfortable 1-0 win over Liverpool in their last-16 second leg, wrapping up a 6-2 aggregate victory.
"If you want to win the competition, you have to be outstanding. We weren't tonight, so that's why it's then fair that we go out," Klopp told reporters.
Liverpool have been eliminated from the League Cup and FA Cup and are sixth in the Premier League on 42 points after a 1-0 defeat at relegation-threatened Bournemouth on Saturday.
"Our job is to squeeze absolutely everything from this season as possible," Klopp said.
"It's a strange one so far, just our recent two games - a sensational performance against Manchester United, a very good football team, and a really bad performance against Bournemouth, again a good football team but we should not lose this game.
"That put us again under more pressure. Three points, and I think everyone would have really felt it, smelt our breath if you like. There is again distance (to the top four) because other teams won their games."
Liverpool faces a tough stretch of games coming up when they travel to Manchester City and Chelsea before hosting leaders Arsenal in the span of nine days.
