Xamarin requires a Mac to compile, but are there remote Mac services out there so I don't need one?

This

http://www.macincloud.com/

might be solution for you, but I did not test it and there are few references about feasibility of this approach and alternatives. MacInCloud has Xamarin Studio on their list and support for public static IP addresses.

Some Xamarin forums discussion about this idea (Mac environment and requirements):

http://forums.xamarin.com/discussion/8345/is-there-a-way-to-build-in-the-cloud

There is more info in the forums, but not much more...