No, a football club does not have a nationality in the legal sense. Nationality applies to natural persons (citizens) or, in some contexts, to corporate entities registered in a specific jurisdiction—but not to sports clubs as such.
FC Barcelona is a multi-sport club based in Barcelona, Catalonia, Spain. It is organized as a societat anònima esportiva (public limited sports company) owned by its members (socis). The club represents the city and region in competitions, but it is not a sovereign entity and does not hold a passport or citizenship.
Players, coaches, and staff have individual nationalities; the club itself has a registration with the Royal Spanish Football Federation (RFEF) and UEFA, which determines its eligibility for domestic and European competitions.