Hola Ferenc

You're right, the verb estar is often used with the adjective "ocupado/a" because it'd have this nuance of expressing a state of mind. See lesson here. 

The reason why we didn't consider that option here (although we will add it) is because the natural way of using estar ocupado is when you say that "a person" is busy, e.g. Estoy muy ocupado esta semana, Estoy tan ocupado que no tengo tiempo para ver a mis amigos... You can also say that you "have" a busy life: Marisa tiene una vida muy ocupada. 

In this case, with "nuestras vidas" as a subject, it's a lot more natural to say "ser" as we did in the exercise, with this expressing an essential characteristic of our lives. 

I hope this clarified it.

Saludos cordiales

Inma