The 38-year-old former Bury and Crawley boss was appointed on Monday to replace Guy Whittingham following his sacking a fortnight ago.
Barker inherits a side 17th in League Two and without a win in eight games, but he is looking to guide them back into League One and beyond.
"I'm extremely proud, it's a fantastic football club with history and potential," he said.
"We need a short-term goal, which is to get into the top half of League Two, and then eventually get out of League Two in the right direction.
"In the long-term, who knows? That is what excited me about coming here. There is no glass ceiling.
"If this place gets rolling it can keep going, the end goal is to keep going.
"I got out of League Two once as a player and once as a caretaker manager so hopefully I can use that experience."
Joining Barker at Fratton Park is Steve Coppell, who will take on the director of football role he occupied alongside Barker at Crawley.
The 58-year-old former Crystal Palace boss said: "This club has a lot of growing to do so it can return to where it belongs in the football hierarchy.
"If I can be a part of that and have a positive input in any department then I'll be very pleased."