Former Macclesfield hitman Matthew Tipton could have given the home side the lead in the first minute, but for a timely block from Anthony Charles.
Ben Strevens went agonisingly close to giving Barnet the opener, but Bury midfielder David Flitcroft made a superb last-ditch tackle to deny him from close range.
Bury keeper Craig Dootson made a splendid stop midway through the first half as Barnet pressured, before on-loan Preston midfielder Joe Anyinsah fired wide.
Shakers boss Chris Casper's side looked to have taken the lead midway through the second half, only for referee Robert Lewis to rule out Jake Speight's lob for offside.
At the other end Barnet scored, but Lewis also deemed the goal to be offside as Guiliano Grazioli netted but to no avail.
Dwayne Mattis fired over the crossbar as the half wore on and Barnet scored another offside goal, this time Tresor Kandol the unlucky player to have his effort cancelled out.
Tipton fired wide as Bury pressed for an opener, but try as they did; there was no way through a resilient Barnet defence.
But, despite a string of chances for both teams, neither could take advantage to win three points and push themselves up the table.