How was the Byzantine Empire able to survive so long, and what were its most important achievements?

The Byzantine Empire was a beneficiary of its
geography. Constantinople's situation on the Golden Horn at the Straits of Bosporus
meant that it was surrounded on three sides by water. The Straits themselves are very
narrow, thus making them very easy to defend. On its land side, a tremendous wall
protected the city. The wall was so wide that a team of horses pulling a chariot could
be turned around on its surface. Those who attacked the city often found themselves
subjected to a flammable and deadly concoction known as Greek Fire, which burned and was
difficult to remove. It is entirely possible that the City of Constantinople would have
survived longer had the soldiers of the Fourth Crusade not forced their way into the
city by artifice and sacked it; from which it never
recovered.

It's greatest achievement was the preservation
of Roman Society and its learning. The Italian Renaissance benefited from scholars from
Byzantium who travelled West after the collapse of the Empire. Constantinople was
considered the "Second Rome" until its collapse in 1453. Additionally, Eastern Orthodox
Christianity originated in Byzantium and from there was transmitted to large portions of
Eastern Europe, including Russia. These areas still practice Eastern Orthodoxy as it was
practiced in Byzantium.