It's not a built-in option of StartOS, but since it allows access over SSH, you can upload/download whatever you want using ssh or scp. You'll have to know your way around linux and the StartOS filesystem structure.

Here is some documentation to migrate the blockchain TO startos: https://docs.start9.com/0.3.5.x/service-guides/bitcoin/blockchain-copy#blockchain-copy

(you'll have to do the reverse when you want to copy it from startos to another machine)