The England international impressed in midfield for the Merseyside club on his Toffees debut at Goodison Park on Saturday and thinks that Roberto Martinez's side might surprise a few people this campaign.
Barry said: "Nobody is that stable and nobody looks like they will run away with the league. That means there could be a surprise this season, you never know.
"And from an Everton point of view, we want to get into the mix and finish as high as we can. Getting into Europe with Everton has to be a target. And the longer the season goes on, the more we'll set ourselves more targets."
One man who looks set to benefit from Barry's season-long loan spell at Goodison is James McCarthy.
McCarthy signed for £13millon on deadline day alongside Manchester City man Barry and he was full of praise for his new team-mate following the win over Jose Mourinho's Chelsea.
McCarthy said: "He is brilliant. He got about, chased Chelsea down and never stopped harassing them. It will be a huge boost for me to play alongside him.
"I've only trained with him a few times but already he's been talking to me a lot, and he's a joy to work with. Obviously I'll learn a lot from training with Gareth. He's a great addition."
McCarthy, who made his Blues bow as a 66th-minute substitute, was thrilled to have been involved in his new club's first Premier League win of the season.
McCarthy added: "It felt like a massive achievement to make my debut at a club like Everton, the gaffer said I'd get a chance and I enjoyed it even if it was a difficult game. I'm just glad we got the points."