You just need root access + a JVM. Once you have that, working with Compojure (and its embedded Jetty) is just like it is on your desktop.
I recommend Rackspace Cloud (http://www.rackspacecloud.com/). It's pay-as-you-go, which is great for testing, and leaving a server running full-time only costs around $12/mo for their small instance. Plus it makes it really easy to test distributed applications. They've got a great admin console and API. (I have no affiliation with Rackspace other than being a satisfied customer).
If you want to use a more robust/mainstream Java server instead of Jetty, you can deploy a WAR bundle of your application to Tomcat, Glassfish, Weblogic, etc. Installing and configuring those is beyond the scope of this question, but deploying a WAR to them is a straightforward, well-documented procedure.
Edit:
As the other commenter mentioned, if you just want to get started playing with Compojure, you can do it easily on your local machine. There's no need to move to a server until you want to start serving pages publicly.