The Shakers conceded three goals in quarter of an hour as Wednesday cruised to their first win under new boss Dave Jones.
"I thought a mad 15 minutes cost us dearly. Defensively, we made it pretty easy for them. Unfortunately, at the minute, we can't stay in games," Barker said.
"We gave Huddersfield a three-goal start on Saturday and you can't come here and give Sheffield Wednesday a four-goal start.
"I think at 4-1 we had the three best chances. I'm not saying we would have won but we could have made it nervous for them. They were a bit nervous for 10 minutes.
"You need a bit of luck at times if little Bury are going to come to big Sheffield Wednesday and get something. At the moment it isn't happening."