Roberto Martinez's side have won seven and lost just once in their opening 14 matches of the campaign and sit fifth in the table going into the weekend's fixtures.
Their crowning glory came in Wednesday's 1-0 win over Manchester United at Old Trafford, with the Red Devils now led by former Everton boss David Moyes.
The victory was the Merseysiders' first at Old Trafford in the Premier League in 21 attempts and Barry admits they are a team brimming with confidence who feel they could break into the top four.
Barry told Sky Sports News: "If we set our targets to the highest point, then that is going to be it.
"Getting into that top four is going to be big ask. If you spoke about Everton before the season, then I don't think people would have been talking about us challenging for that spot.
"But the way the first 14 games have gone, I don't see why we cannot set our targets that high.
"It reminds me of the first few months when I joined Man City. That was a changing period, similar to here with the new manager.
"It's his (Martinez's) first season, he is still putting his ideas across and the lads have adapted to them really quickly.
"It's still the first season and you are still learning, but if we can keep improving, then it's going to be brilliant for Everton."