Barry completed a permanent switch to the Merseyside club earlier this week, having spent last season on loan from Manchester City.
The 33-year-old will be expected to lend experience to the likes of Ross Barkley and James McCarthy as Everton go about trying to improve on last season's fifth-place Premier League finish.
"I felt really good last year and I didn't feel too old," he said.
"It sometimes helps playing alongside younger players and we have a great group of them.
"You want to take each season at a time but I don't feel like I am slowing down.
"Me and James had a great season in terms of staying fit, playing a lot of matches together and understanding each other's game.
"That was on the back of a last-minute decision to be brought in here, so I'm hopeful a full pre-season with the team can further improve myself, James and the whole squad together."
