Why was Justinian’s attempt to restore the Roman Empire doomed to failure?

It was not so and it was already a failure. Justinian was able to return Illyria, the territories of North Africa (by defeating the Vandals), and the lands of Italy (by defeating the Ostrogoths). Thus, the eastern Roman Empire was able to resume full control over the Mediterranean. Of course, Justinian was not able to fully return all the lost territories, such as Gaul, Spain and other lands, but still the heyday of Byzantium was under him. True, it is worth considering the fact that at that time the barbarian tribes were much better united and strong than in the days of Rome.