"We're absolutely not too good to go down, nobody's too good to go down," the 23-year-old told BBC Radio Suffolk.
"But if we do justice to ourselves and play the way we can play, do what the manager wants to do, we will be fine."
New manager Paul Jewell's first piece of transfer business at Portman Road was to extend O'Dea's loan stay from Celtic until the end of the season.
The Republic of Ireland international, who has 15 Ipswich starts to his name this season, admits a shared sense of responsibility led to him staying.
"That's partly one of the reasons I wanted to stay, I feel I've been part of the club the last six months," he said.
"I'm fully up for the task ahead and I think everyone's in the positive mind we'll move forwards rather than backwards."
Jewell's first game in charge on Saturday, which saw the Blues beaten 2-1 at Millwall, featured a heated conversation between goalkeeper Marton Fulop and defender Mark Kennedy.
The two collided after Hungarian Fulop pushed a cross behind for a corner, but the shot stopper insists it has been put behind them.
"It's forgotten, I'm quite a positive guy, I like positive feedback from the lads," he said.
"And I always give positive feedback, but the matter is closed. It's not a big deal, you just have to deal with it the right way."
Fulop also backed his compatriot Tamas Priskin to continue his good form in front of goal.
The 24-year-old striker has found the net seven times in 26 games this season, but has scored in each of his last two outings.
"I think he's on a good road now, he's obviously a natural goalscorer," said Fulop.
"When he gets the ball in the box his first touch is very good and his finish is too.
"I don't want to put pressure on him but I know he's capable of scoring more goals."
Source: BBC Sport
Source: BBC Sport