The Iron are currently sixth in the National League, one point and one place outside the play-off spots.
"I thought we asked a lot of questions of a very good Barrow team," Cowley told BBC Essex.
"When you come off the back of a six or seven-hour journey and you concede a goal after five minutes it's always going to be tough."
Barrow extended their unbeaten league run to nine games with the victory, and although Braintree's losing run extended to three games, Cowley remained positive.
"What we've done so far has been nothing short of remarkable," he continued. "But if anyone thought this journey was going to be easy, then they're living in cloud cuckoo land.
"We're a part-time team. This is our toughest challenge to date, but we're going to embrace it. We're going to enjoy it and see where it takes us."
Braintree manager Danny Cowley said the task ahead of his team is "a brilliant challenge" after the defeat at Barrow.