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.
The word "Ortho" means staright. It never deviated from its path. It recieved beating but never bent. Recieved pressure but never caved.