I have been reading history for years, and nowhere have I come across a large civilization that has successfully navigated resource depletion. But if you look at things on a local level, then it becomes pretty clear that people will survive. What is lost is that thin veneer of civilization. History is replete, on every continent, with the rise and fall of various empires and societies.

Things are different today because of a single thing - petroleum. This resource has been the enabler of every single device we consider "modern technology". It has been the enabler of our world's first global society. We are all aware, instantaneously aware, of what transpires anywhere on earth for the first time in history.

It's not just Peak Oil, it's the whole overshoot dilemma beginning to catch up with us - PO is just the first to grab our modern way of life by the throat.

Japan survived an approaching deforestation crisis in the 17th century.  The authoritarian Shogunate implemented strictly enforced rationing and strong centralized control over reforestation.  It's covered in Jared Diamond's book.  The equivalent events in 17th century Europe led to the coal-based industrial revolution, but Japan managed to navigate the transition without doing that, and only industrialized much later when faced with external competition from Europeans.
Well, Athens had another solution for the problem of deforestation of Attica. Intensive imperialism in order to import wood from the Black Sea. They needed the wood to produce energy for the silver mines.
Japan managed to industrialize because they had coal. The coal deposits around Nagasaki were known before the Meiji restoration. They never had as much coal as the British but enough to start. Besides Japan did expand to North, to Hokkaido, where there was large forests. Later they seized the forests of Southern Sakhalin, and Taiwan (with some coal and forests). All these resources could be reached by sea.

But of course we can see the scarcity of wood in the Japanese building and life style.