Jennison Myrie-Williams should have put Vale ahead in the first half before goalkeeper Chris Neal saved twice from Barnet's Andy Yiadom before the break.
The Vale stopper then denied John Oster and, despite a late rally, neither Doug Loft nor Tom Pope could beat Graham Stack as the game finished goalless.
And Adams, whose side saw second-placed Gillingham close the gap with a win at Bradford, admitted his side were second best to Barnet.
"It was a good point based on our performance. Barnet were fantastic, they were better than us," said Adams.
"We didn't have our usual spark, but we could've nicked it at times too. When you consider most of our best players were defenders, that says a lot about our performance.
"We gave Barnet too much time and respect on the ball. Edgar Davids ran the show at times along with (John) Oster. You can't give any type of player at this level that amount of time and space on the ball. We were very, very luck to come in at half-time 0-0.
"We're not going to have it easy. Teams aren't going to roll over and let us tickle their bellies, because that's not how it is at this level. Barnet didn't set up to stop us, they just played their own way. We might have to look at the way teams play when we're away from home."
- Sports & Recreation
- John Oster