I know that the Azure Storage entities (blobs, tables and queues) have a built-in resiliency, meaning that they are replicated to 3 different servers in the same datacenter.
Depending on where you want to backup your data, there are two options available:
Backing up data locally - If you wish to backup your data locally in your infrastructure, you could: a. Write your own application using either Storage Client Library or consuming REST API or b. Use 3rd party tools like Cerebrata Azure Management Cmdlets (Disclosure: I work for Cerebrata).
Backing up data in the cloud - Recently, Windows Azure Storage team announced Asynchronous Copy Blob functionality which will essentially allow you to copy data from one storage account to another storage account without downloading the data locally. The catch here is that your target storage account should be created after 7th June 2012. You can read more about this functionality on Windows Azure Blog: http://blogs.msdn.com/b/windowsazurestorage/archive/2012/06/12/introducing-asynchronous-cross-account-copy-blob.aspx.
Hope this helps.