The Bees enjoyed the greater possession – and chances in the first half but couldn't press home their advantage.
Captain Neal Bishop probably should have headed them into the lead in just the 12th minute, though unmarked at the far post from Tommy Black's free-kick, he headed wide.
Luke Daniels in the visitors' goal was then called into action when he clawed away John O'Flynn's cross, before Nicky Deverdics flashed a shot wide of the far post.
The Shrews did have a couple of noteworthy chances, both falling to top goalscorer Grant Holt.
First he was presented with a golden opportunity from just a matter of yards out but the striker couldn't get the ball out of his feet, before soon after he only had Ran Kadoch in the home goal to beat, but skewed his shot wide.
Barnet came within inches of taking the lead in the last minute of the half though O'Flynn saw his effort smack back off the bar.
The visitors seemed to come out with more purpose after the break with Steven Leslie forcing a fine stop from Kadoch. Albert Adomah then stung the hands of Daniels with a low, fierce strike which the keeper fumbled, though as the half wore on chances were becoming at a premium.
Neither side created anything of note in the last half-hour of the match and although it was a frustrating encounter, a-point-a-piece was all that either side deserved.