You can now use the Azure CDN to access blobs with custom domains over HTTPS. See the following article for instructions on how to do so: https://docs.microsoft.com/en-us/azure/storage/storage-https-custom-domain-cdn. Having talked to a number of customers, we concluded that this solution addresses many scenarios where the need for HTTPS access to blobs with custom domains exists.
Native Azure Storage support for using SSL to access blobs at custom domains is still on our backlog. We would love to hear about your scenarios where using the Azure CDN is not an acceptable solution, either by posting on this thread or sending us an email at email@example.com.Michael commented
You can realize this with Azure Functions Proxies that route to the blob storage. Just add your custom domain to the Function App. The Proxies defined there route to the blob storage like this:
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.Michael commented
You can realize this with Azure Functions Proxies that route to the blob storage.