One of the most popular requests for Azure storage is to host a web site in a blob, and they do say that they will get around to it this year.
In the meantime they explain you you can get your domain name hooked up to a blob storage end point
but let’s face it who wants to tell their users to type in a file name after the domain name in their url?
Recipe
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Create a storage account with a container (named docs for exampe) with access policy Blob
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Create a CName record for a subdomain
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Upload your files to the container
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Create an Http redirect record to the file at the subdomain ( remember to use the container name)
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Tell Azure to use the subdomain
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Celebrate ( if all went well )
If all did not go well try checking whether the dns records are resolving using https://www.digwebinterface.com/ with Trace