A drop-in solution to achieve the same is using CloudFlare as CDN for your website. CloudFlare has a configurable "Always Online" mode that is automatically triggered whenever your site is down, that shows the user an offline version of the website, together with a warning message.

Obviously, if you're using CloudFlare in the first place, chances are that you won't have too many problems with high peaks of traffic anyway. But obviously, it can still happen, depending on how high the peak is :)

https://support.cloudflare.com/entries/22050652-What-does-Al...