I would like to be able to use Google Desktop Search to search data that is stored on other computers, as well as my desktop - specifically, data that's stored on a Sharepoint installation on my network. Is it possible to get the Sharepoint install indexed?
Is it also possible to store this index in a central location, and share it with other people that may join the network?
If you map the network drive from the server running Sharepoint to your local machine, then yes.
Otherwise, no: Google Desktop is for searching your desktop - not the network.
You should look at something like the Google Search Appliances or Mini to meet what you're describing.