> Lots of the early archeology exported antiquities legally. At the time most of the Middle East, Greece included, was ruled by the Ottoman Empire that granted official permissions for western archeologists to take objects or even whole buildings and edifices.

But this answer can't satisfy anyone who cares about the issue, because there's no good argument that these exports were legitimate, just legal. Compare the Soviet Union's early efforts to sell off treasure and art belonging to the Czar -- they couldn't do it, because no one would buy their stolen goods. (Eventually, this broke down as the Soviets stayed in power.)

Saying it's fine for the British Museum to have Greek antiquities because the Ottoman Empire, the mortal enemies of the Greeks, gave them permission to take whatever they wanted... looks ridiculous. Imagine China freely granting permission to foreign archaeologists to dismantle and export Tibetan palaces and monasteries.