Did ABBA have a backup band?

Yes. In fact the ABBA sound owes much to their backing musicicans, who were some of the best in Europe. Regulars, who appeared on most of ABBA's albums and tours, included:

Rutger Gunnarsson - Bass

Janne Schaffer - Guitar

Lasse Wellander - Guitar

Ola Brunkert - Drums

Roger Palm - Drums