Can you create a clustered DB in Azure using Windows 2003 and SQL Server 2005?
Shared Disks are currently in preview. https://docs.microsoft.com/en-us/azure/virtual-machines/windows/disks-shared

But if you have no choice but to run old, unsupported software, you shouldn't attempt anything as complicated to implement and operate as cluster.  Just install the software on a single VM with premium disks, and live with the 99.9% uptime SLA for a single Azure VM.  And with a single stand-alone VM, you may have an easier time supporting a more current OS or SQL Server version.

Put another way, a failover cluster on with Windows Server 2003 and SQL Server 2005 will definitely be more expensive, and will probably have worse uptime than a single standalone VM.  With a cluster you have to properly configure and be able to troubleshoot the quorum model, the Azure load balancer that supports the floating IP, and you must have a domain with highly-available domain controllers.   The likelihood that you get all this right, and have the ability to troubleshoot and repair it if it gets messed up is just too low.  And if you can't figure something out, you won't be able to get help.