what works out cheaper in barcelona...a hotel or self catering? for a family of 2 ad and 2 children?

Self catering will be cheaper, you can eat very cheaply in the tapas bars and the restaurants doing menu del dia and you will have more freedom for the children.

Self catering, going to the supermarket and the market and cook yourself.
But do you want to do the household cores when on holiday?
A good compromise may be eating out for lunch, when the meals are cheaper, and having bread and local meats, cheese or jams for dinner (or whatever the kids like best).
Might be better with the children too, dinner in Spain can be rather late, and if your children are not used to it, they will be rather tired before dinner is served.