UPDATE: At the time of this answer the Azure Portal did not have this feature. It does now as outlined here. The following outlines the way to do this before the UI was added.

How can I configure CORS in Azure BLOB Storage in Portal?

If you want you can always set the CORS rules for blob storage programmatically. If you’re using .Net Storage Client library, check out this blog post from storage team: http://blogs.msdn.com/b/windowsazurestorage/archive/2014/02/03/windows-azure-storage-introducing-cors.aspx. Code for setting CORS setting from that blog post:

private static void InitializeCors()
{
     // CORS should be enabled once at service startup
     // Given a BlobClient, download the current Service Properties 
     ServiceProperties blobServiceProperties = BlobClient.GetServiceProperties();
     ServiceProperties tableServiceProperties = TableClient.GetServiceProperties();

     // Enable and Configure CORS
     ConfigureCors(blobServiceProperties);
     ConfigureCors(tableServiceProperties);

     // Commit the CORS changes into the Service Properties
     BlobClient.SetServiceProperties(blobServiceProperties);
     TableClient.SetServiceProperties(tableServiceProperties);
}

private static void ConfigureCors(ServiceProperties serviceProperties)
{
    serviceProperties.Cors = new CorsProperties();
    serviceProperties.Cors.CorsRules.Add(new CorsRule()
    {
        AllowedHeaders = new List<string>() { "*" },
        AllowedMethods = CorsHttpMethods.Put | CorsHttpMethods.Get | CorsHttpMethods.Head | CorsHttpMethods.Post,
        AllowedOrigins = new List<string>() { "*" },
        ExposedHeaders = new List<string>() { "*" },
        MaxAgeInSeconds = 1800 // 30 minutes
     });
}

If you’re looking for a tool to do the same, a few storage explorers have support for configuring CORS – Azure Storage Explorer, Cerebrata Azure Management Studio, Cloud Portam (Disclosure – I’m building Cloud Portam utility).

Once the CORS is configured properly, you can use the code mentioned in Rory’s answer to download the file from blob storage. You don’t have to do anything special on the client side as mentioned by Rory.