Is there any evidence that humans have faced extinction before?
There is ample genetic evidence for a population bottleneck following the out-of-Africa migration. This would account for the reduced genetic diversity found in non-African populations. 

There is further evidence for an earlier major bottleneck that reduced the human population to around 10,000 individuals.

These kinds of studies are typically done now via Coalescent theory using whole-genome sequences.



I just found this nice blog post summarizing the Nature paper I linked to.