The hosts controlled much of the first half and created enough opportunities to take a commanding lead, but Celta goalkeeper Radu proved difficult to beat.
Jon Martin came closest early on when his powerful header forced Radu into a strong save. Mikel Oyarzabal then tested the Romanian goalkeeper with a fierce left-footed effort before Takefusa Kubo somehow fired over from close range with the goal seemingly at his mercy.
Sociedad continued to apply pressure, with Yangel Herrera forcing Radu into another save before Benat Turrientes saw his 25-yard strike stopped by the Celta goalkeeper.
Despite their dominance before the break, Sociedad could not maintain the same intensity in the second half.
Celta began to threaten on the counter-attack, with Andres Antanon wasting a promising opportunity when he fired wide before the game became increasingly physical.
Yoel Lago and former Chelsea defender Marcos Alonso were both booked as the visitors looked to disrupt Sociedad's rhythm.
The home side believed they should have been awarded a penalty with less than 15 minutes remaining when Goncalo Guedes went down inside the area after appearing to be pulled back by Alvaro Nunez. Referee VÃctor GarcÃa Verdura dismissed the appeals.
Celta then came close to snatching all three points when veteran forward Iago Aspas tested Sociedad goalkeeper Remiro, who made a crucial save to keep the scores level.
The result marked the first goalless draw between the clubs since November 2008, leaving Sociedad frustrated after failing to turn their first-half dominance into victory.
Real Sociedad now travel to Elche on Monday, while Celta Vigo face another away trip against Getafe on the same day.