Hodgson's side suffered 2-1 defeats to both Italy and Uruguay before a goalless draw against Costa Rica saw them finish bottom of their group.
The England manager insists the squad's preparations for the tournament were encouraging, but he felt they suffered from bad fortune rather than form.
Asked for his reflections on the World Cup almost two months on, he said: "Still great disappointment, and the feeling of enormous frustration because the preparations had gone very well.
"I thought the players were very focused and everyone was so committed to doing very well and we honestly believed we could.
"We lose the first game and in my opinion, a little bit unluckily. We certainly played well enough and deserved more. Sadly we go into the second game and things are going okay and we got ourselves going.
"In the second half, we see a bad goal conceded and somewhat a freakish goal and suddenly you've lost two and you're out of the competition before we've even started. It was a bitter blow and it's going to take more than a couple of months to come out of that.
“We have a lot of rebuilding to do not only of rebuilding our team but also rebuilding all the faith that people showed us. We felt that people were behind us and like ourselves believing this could be a good tournament for us but it turns out to be a bad tournament for you.
“As a result, we have to look forward to 2016 and make certain that we don’t allow it to happen again and hope we don’t have the same lack of fortune as well that we perhaps suffered as well.”
Steven Gerrard captained England in Brazil and announced his retirement from international football last month, but Hodgson insisted he never had any intention of standing down.
“There has been plenty of dark moments plenty of times after looking back on things that you can’t believe fate has put you in that sort of situation,” he added.
“The fact is I would quite happily relinquish this job when my employers think don’t want me here anymore and the players don’t have the respect necessary and that they don’t want to continue to play or me. In that moment if it is time for me to walk away, I will do so.
“But this time, it was made clear to me that that wasn’t the situation that the FA wanted me to continue in the job and the feedback within the team was the same. So therefore there was never any doubt in my mind that I shouldn’t continue.
“You only walk away when, A, you can’t stand the pressure or, B, that you don’t feel that you are good enough to do the job and in that situation, I can safely and strongly say that I never felt that at all.”
Hodgson also played down suggestions that England should play with their own identity, insisting his side’s style compared favourably with World Cup winners Germany.
He said: “I don’t believe in the nonsense spoken about the identity of certain countries and how they play. I don’t see the vast amount of difference between how we try and the way Germany try to play.
“Now there is a major difference between the 11 players in the German shirt and the 11 players in the England shirt. But if we are talking about how they attack and build up their attacks, or how they keep possession and seek for openings when they have got the ball, I don’t see a vast amount of difference in philosophy or style.
“And the same defensively. I don’t see what they are trying to do to keep compact to put pressure on the ball, to try and keep the team united close together, to block crosses. I don’t see the great difference in philosophy there either.
“If ever you want to know about the small margins in football and the fact that it is anything other than an exact science, that random things that happen in the game makes all the difference, I could give you six or seven examples from the World Cup that made winners into losers and vice-versa.”
Hodgson also paid tribute to Gerrard and revealed he wanted the Liverpool midfielder to retain the armband for the forthcoming European Championship campaign.
Asked if he expected Gerrard to quit international football, Hodgson replied: “No not necessarily. It was a tough decision for him and one that he agonised over.
“It would have been easy for him after the World Cup just to say ‘well, it has been tremendous, I‘m disappointed it hasn’t gone better for us but that’s it for me'.
“But he didn’t do that and the reason why is because he wanted to consider all of the things we talked about and make a decision he could accept and live with. It wasn’t easy for him to turn his back on England because he so much enjoyed playing for England and he’s done such a good job for us.
“I went through all the possible discussion points that you could have and I made my points of why I wanted him to stay on and he promised to give it a thought. I promised him that I know he would have given a lot of thought at all potential points that I made, and so I will accept with good grace whatever his decision is.”
Frank Lampard was England’s vice-captain in Brazil despite not featuring in the first two games, while Wayne Rooney is widely regarded as the favourite to take over the captaincy from Gerrard.
“Frank is the vice-captain and there is Wayne Rooney so there are two candidates,” said Hodgson, who was then asked if he knew who the next captain will be and whether the new skipper had been informed.
“Yes to the first part and no to the second,” he added. “We don’t need to compare to other European countries, let’s just stay as England. I have got a big job ahead of me I believe to choose that captain and that’s what I’m working on.
“I have half chosen him. The person who is captain for the Norway game will be the person I will be designating to hopefully be the captain in 2016.
“At the moment, Frank’s concentration has to be on this Manchester City job and in New York, where he could very well be captain.
“I don’t think I can expect him to say ‘please make me captain because in 2018 I want to lead the team out’, because I don’t think he will be thinking that himself.
“But having said that, if he is with us in the first game, who knows. Then he could captain us that day.”
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England play Norway at Wembley on Wednesday 3 September, tickets are under £30 and available at the FA’s website www.thefa.com/tickets