What is a good strategy to keep my site online when S3 goes offline?
In March 2015, Amazon AWS announced they support S3 replication across regions. When a certain region in S3 goes offline, you can serve files from your mirror in another region.

source: https://aws.amazon.com/blogs/aws/new-cross-region-replication-for-amazon-s3/

The practice of keeping your infrastructure online by doing a switch over to another region is a complex one, but S3 is a relatively small and simple component. Netflix has a great article on their experience with Chaos Gorilla.

This also applies to service degradation, like increased latency. Not just when a service you depend upon is completely offline. Netflix has an article on this as well: Chaos Engineering Upgraded.