This is why Jews are Jews and not Christians, among many other reasons, but I don't think I ever really learned the historical reasoning for the rejection at the time of his death. Whether you believe Jesus is the son of God, he was a real person.
I just don't understand how and why exactly they didn't believe he was the son of God, and how they managed to survive for so long not believing it.
It didn't. That's why Constantinople is called Istanbul.
It survived for more than 1100 years after Constantine. Was there ever an empire that lived longer?
Wasn't really an empire for that entire time. Byzantine never became the behemoth the Roman Empire was. And it was destroyed for pretty much the same reasons as Rome. Inner religious conflicts weakened it to a point where it couldn't defend itself anymore from the Muslim invaders.