The Blaugrana threw away an early two-goal lead and then a late one-goal advantage in a result that could spell the end of their title challenge.
The draw left Barca one point behind first-place Atletico Madrid, who now have a game in hand over Koeman's side as the season winds down.
What was said?
"As a coach I am the one most responsible," Koeman said at his post-game press conference. "I am asking myself what we did at half-time, because we had talked about continuing with the intensity and improving things defensively. Watching the second half, that didn't happen. We have lowered the intensity.
"The players, technical staff and coach, we are all sad.
"Now, we have to prepare for the last two games and prepare ourselves for the next season. But first we have to get over today's draw, which is a hard thing to do.
"We entered the second half with less intensity, reacting late to all plays. We have gone down in everything and it has been a problem."
The bigger picture
In an extremely tight Liga race, Barcelona could have taken over first place had they held on against Levante on Tuesday.
Instead, Sergio Leon's 83rd-minute equalizer consigned Barca to long-shot status in the Spanish top flight.
Barca have just two games remaining while Atletico, as well as third-place Real Madrid who are a point behind the Blaugrana, have three matches left.