Yet despite failing to win at home for the fifth match in succession, Reading remain sixth in the table.
"On the back of a good win away to Birmingham, that was a great opportunity for us tonight," Adkins said.
"We were playing a team well below us in the league and yet we failed to capitalise on that. Across the team, we were poor. We were just not good enough."
Reading were mocked at the final whistle by many of their fans, who chanted 'That was embarrassing'.
Adkins as good as agreed with them, saying: "The fans can go and do that. I was probably one of them.
"We're still in the play-off race but we were very poor this evening. No one can hide away from that fact."
Reading, unbeaten in their previous five matches, started sluggishly and allowed Barnsley to take the lead in the 16th minute through a poor defensive error.
Chris Gunter's backpass fell woefully short of keeper Alex McCarthy and Chris O'Grady nipped in to cleverly set up skipper Stephen Dawson, who beat McCarthy with a well-guided shot.
Reading levelled fortuitously, when Pavel Pogrebnyak was needlessly brought down by Kelvin Etuhu as he chased a long pass from Jobi McAnuff.
Pogrebnyak coolly converted from the spot for his 10th goal of the season.
Barnsley continued to battle gamely, dominating their higher-placed opponents, and could have regained their lead before the interval, but O'Grady drove wide from a good opening.
However, they made no mistake in the early stages of the second period, scoring twice in the space of four minutes to effectively clinch only their second away win of the season.
In the 53rd minute, Tomasz Cywka flicked on the ball for Reuben Noble-Lazarus, a half-time substitute, who curled in a fine shot over McCarthy.
Soon after, Reading were stunned again. Dale Jennings cut in unchallenged from the left flank and unleashed an unstoppable 25-yard drive past McCarthy.
Reading huffed and puffed from then on, with Royston Drenthe twice going close and Alex Pearce forcing a good save from keeper Luke Steele, but Barnsley held on comfortably.
Danny Wilson's team moved off the bottom of the table and into 23rd place, leapfrogging Millwall on goal difference.
"Of course we can stay up," Tykes boss Wilson said. "When you witness a performance like that, we must have a great chance.
"We fully deserved the win and it will give our confidence a great boost. All in all, it was a superb performance.
"Our energy levels were fantastic, especially in the way we pressed Reading, and we got everything we merited.
"Even when they had equalised, we rallied and never let it distract us. We quickly got back on the horse.
"Our second and third goals were terrific, two bits of quality against any team."