Roy Hodgson's men were on the brink of elimination in Group D after losing 2-1 to Uruguay in their second game on Thursday and England's worst fears were confirmed when Costa Rica stunned Italy 1-0 on Friday.
England's failure marks the first time they have been unable to make it past the first round at a World Cup since 1958, and Chelsea centre-back Cahill feels despondent due to their performance in Brazil.
"A few weeks ago it was one of the highlights of my career and a couple of weeks down the line it's one of the lowest points," the 28-year-old said.
"It's the worst feeling I've had as a player. There's nothing more I can say. It's very tough to take, football can be cruel sometimes."
Cahill added: "Things will be written, things will be said.
"But there's nothing you can do about it. I think we will take the full responsibility for what happened."
England round off their World Cup campaign against Costa Rica, who have already qualified for the last 16, in Belo Horizonte on Tuesday.
