Thank you for your feedback. We are currently in public preview of static website hosting for Azure Storage to enable this scenario. Check out the blog post here for more details: https://azure.microsoft.com/en-us/blog/azure-storage-static-web-hosting-public-preview. The feature set includes support for default documents and custom error documents for HTTP status code 404.
For any further questions, or to discuss your specific scenario, send us an email at firstname.lastname@example.org.Peter Reid commented
A custom 404 handler for Blob Storage would enable me to build a "content pull" model for the CDN. A request for content from the CDN goes back to Blob storage, if its not there then a custom 404 handler will grab the content from the source content system. Very powerful feature