In some ways it is miraculous the Eastern Roman Empire lasted so long. Steven Runciman put it well:

In some ways it is miraculous the Eastern Roman Empire lasted so long. Steven Runciman put it well: “Few states have been organized in a manner so well suited to the times and so carefully directed to prevent power remaining in the hands of the incompetent...fundamentally it was a heritage from the Roman past, but continually it had been adapted and supplemented throughout the centuries to suit their varying requirements.”

It is easy to identify some incompetent Emperors, of course, over a period spanning over a millennium. But, the position of the Emperor was in fact quite vulnerable. It required support from three areas of society to be fully secure in rule: the army, the Senate (essentially the aristocracy and officeholders receiving a salary from the state), and the people of Constantinople. Runciman said that “though there was no constitutional check on his power, the Emperors autocracy was nonetheless limited. He always recognized his obligation to respect the fundamental laws of the Roman people; and deep down, there lingered the idea that the sovereignty was the people’s, and the people had only delegated power to the Emperor.” This delegation is exactly the sentiment Augustus cultivated. 

Of course, the “Byzantines” didn’t truly have any thing actually like a republic - but they did have a de facto checks and balances. Because if the Emperor went too far, the army, aristocracy, the people, or most dangerously a combination of these elements, could overthrow him. This is why Emperors were always desperate for legitimacy if they were lacking it - their life was on the line.

Source - Byzantine Civilization by Steven Runciman