The 31-year-old 'keeper got his fingertips to Gornell's driven long-range effort which seemed destined for the top corner.
Jason Kennedy almost snatched a late winner for the visitors with a well-struck half-volley which sailed less than a yard over the crossbar, with Stack rooted to the ground.
Barnet's on-loan debutant Dani Lopez, who scored six goals in a 12-game stint at Aldershot earlier this season, threatened to open the scoring on several occasions.
The 27-year-old Spaniard, on an initial month-long deal from Stevenage, headed wide from Mark Byrne's 58th-minute delivery, shortly after he had nearly scooped the ball into the net from a corner.
Lopez almost opened the scoring in the first half after his fancy footwork beat Rhys Bennett only for his goalbound effort to be deflected wide.
Gornell had earlier headed Andrew Tutte's 19th-minute cross wide of Stack's goal despite being unmarked from 12 yards in a first half of limited opportunities.