The 31-year-old, who joined the Bees on a free transfer in 2012, had been expected to depart following their relegation from League Two.
However, the former Arsenal shot-stopper told Sky Sports News: "I signed a three-and-a-half-year deal in January so I showed my commitment. I am not one to shy from any challenge. I have got friends who play in the Conference.
"You have only got to look at Luton sitting in mid-table. But Barnet is a big club and we have got enough quality within the group now. The big question is who is going to be here next season? The club will have to restructure financially with wages.
"If we get a group together who are determined, I can't see there being a problem. But it is going to be a tough, tough ask. It will be a massive effort for us get back in the Football League but it is a challenge I am sure everyone is relishing."
Stack has also previously played for Hibernian.