What does it mean to have a spiritual body? Do the bodies of Radha and Krishna have excretory systems, other organs, etc?
Radha and Krishna have bodies ( vigraha) that are created of sat ("eternity"), chit ("knowledge") and bliss, ananda. Their bodies are fully spiritual. They do not have systems and organs like the biological, material body. They never get old, ill, nor are They forced to die. This is also the situation with the jiva, or individual soul; we too, in our eternal spiritual nature, are eternal and never die.
You can read more about the nature of the soul in the 2nd chapter of Bhagavad-gita. There's also some information about Krishna's sat, chit, ananda nature in the introduction of the Bhagavad Gita As It Is, by Srila Prabhupada.
I hope this is helpful.