The Gunners were downed 4-1 on the night by a Lionel Messi four goal salvo, however they had threatened to overturn their 3-2 first leg deficit when Nicklas Bendtner struck early in the 19th minute.
But Barca outclassed Arsene Wenger's team and quickly stamped out any prospect of a dramatic turnaround, with the result leaving the London club with just the Premier League to play for.
Even that maybe beyond Wenger's team as Chelsea sit three points clear the division's summit, and Campbell - re-signed by Arsenal in January - says his younger colleagues must do some soul-searching.
"Everyone here has to learn and be honest about it," the 35-year-old said in the News of the World. "To ask where they want their careers going.
"Say that this is the level I want to get to and surpass. But you have to be honest with yourself and say 'I've got to work harder, I've got to understand the game a bit better and realise it is a big world out there and there are a lot of good players out there at big clubs'.
"This is where you learn. You learn a lot about yourself and where you are going to be. You can't think you're okay and you're not. You've got to be a very quick learner.
"This is the level you have got to be at. You've got to look at yourself. This is a different level but you have got to appreciate what they are doing and raise your level."
Arsenal have gone five years without a trophy, their last success coming in a 2005 FA Cup trophy final, but manager Wenger has refused to take a pragmatic approach to silverware.
Instead, the Frenchman has looked to gradually construct a team capable of emulating the famed 'Invincibles' of the 2004 season, where the Gunners won the Premier League title by going unbeaten all season.
Campbell's team are still in the hunt to take the crown this season, although it remains in their rivals hands.
And the former England international, who started the season with League Two outfit Notts County, believes Arsenal must unleash their potential and start delivering trophies following the barren run.
On the comparisons between the current team and the unbeaten squad he was a part of, Campbell added: "They are two different periods - there were some special guys back then.
"That's not to say this new set of players is not that. They are fantastic - they have got great talent. But it's getting the balance right. It's all about winning something - going for it every year with all guns blazing.
"You've got to respect football in every way. You've got to go for the League Cup, you've got to go for the FA Cup, you've got to go for all competitions because you never know which one is going to come your way."