Barca play Real Sociedad, Real Madrid face Celta Vigo and Atletico Madrid meet Eibar in three ties involving two La Liga teams.
Alcoron, the only non top-flight team left, have been drawn against Alaves.
The first legs will take place on Wednesday and Thursday, with the return fixtures a week later.
Barca, who fought back from a first-leg deficit to beat Athletic Bilbao in the last 16, are aiming to become the first team since 1953 to win three consecutive Copa del Reys.
Full draw:
Real Sociedad v Barcelona
Alcorcon v Alaves
Atletico Madrid v Eibar
Real Madrid v Celta
Holders Barcelona have avoided Real Madrid and Atletico Madrid as Spain's big three clubs were kept apart in the Copa del Rey quarter-final draw.