What are the Linux distributions that are compatible with the Immutable repository?
Can you give me some advice on the best hardware or virtual machine configurations to deploy a large backup repository to hold all of my virtual machine backups that will be kept for a year?
According to https://www.veeam.com/blog/installing-ubuntu-linux-veeam-hardened-repository.html, Ubuntu Server Linux is an example.
What about a single large virtual machine, a LUN formatted with ExtFS, or some other fancy secure file system?
Since you are planning to use it with Veeam B&R, I would highly recommend you follow their best practices and recommendations where Ubuntu, Red Hat, and SUSE are mentioned: https://www.veeam.com/blog/hardened-linux-repository-best-practices.html
An excellent option is using a pre-built and pre-configured Veeam Hardened Repository Appliance https://www.starwindsoftware.com/blog/starwind-san-nas-as-hardened-repository-for-veeam-br, which already has all the essential things configured in accordance with the above practices, can be deployed bare-metal or as a virtual machine and is free to use.