The Eastern Roman Empire had invaders. Take note of the Visigoths in the 4th century AD, the Ostrogoths in the 5th century (who the Eastern Romans were having many problems with before they eventually invaded Italy), and also take note of the Slavs who invaded the Balkans, including Greece, during the 6th and 7th centuries. And let's not forget that the rise of Islam took from the Eastern Romans many of their most valuable provinces, and later after 1071, the Seljuk Turks.
The big reason why the Eastern Roman Empire outlasted the Western Roman Empire by 1000 years was simply due to more effective leadership. The Eastern Roman Empire was much more densely populated and urbanized than the Western Roman Empire - hence collecting taxes on the people and enacting policies which all individuals must follow as well was much easier in the eastern empire (because it's much easier to do these things in cities than a vast countryside sprawled with tiny villages here and there with no real administrative center for hundreds of miles). This was one of the reasons why the Western Roman Empire couldn't stand the barbarian invasions and thus why they had to rely on the barbarians to protect the borders, which in the end turned on itself.
The wealth of the eastern empire was also much greater. By about 300 AD, the western empire had become sort of a backwater, even in Italy (though some cities were still fairly well off, Milan being one example; though those were few and far between). Most of the population and the bulk of economic activity was concentrated in the eastern Mediterranean region. The division of the empire by East and West left the Western Roman Empire having to control too much land with too little of a tax base, unlike the Eastern Roman Empire. In the end, it was not really sustainable.
The same fate would occur on the Eastern Roman Empire about 1000 years later also, though there is a difference. The fall of the Eastern Roman Empire was clear while the fall of the Western Roman Empire was not and this is largely because we need to keep in mind that the eastern empire was based around its capital of Constantinople whereas the western empire had no real focus - even Rome during the Late Antiquity was nothing compared to its past. The Eastern Roman Empire over its last centuries, like the western empire, had lost much land and wealth. Its capital of Constantinople, however, was still relatively well off despite everything but alone against the Turks who now possessed cannons that could fire through its high walls, it could not stand a chance.