The point, only their fourth from a possible 60, kept them above bottom-club Pools in the League One table.
Pools beat Pompey at Fratton Park just two weeks ago but they failed to find a way through a packed and effective rearguard despite endless pressure in the second half.
Pompey's Hartlepool-born winger Adam Reed had the game's first chance, shooting low across goal after seven minutes.
The home side, for all their pressure, had to wait until the 43rd minute to test goalkeeper Simon Eastwood, when Charlie Wyke fired low.
Pools midfielder Jack Baldwin's burst into the area saw him fire low across the face of goal with Eastwood beaten, before his low 20-yard snap shot fizzed wide.
James Poole had two late chances for the home side but saw his efforts charged down.
At the other end, Reed's counter-attack could have brought an injury-time winner but Peter Hartley cut out the shot.
Pools made it four games without defeat but they remain 11 points shy of safety at the bottom.