Is it possible to use the EE Classes within a template?
Are they only available for add-on development?
You'd need to create an add-on to use them, but creating a plugin is simple enough and if you needed something beefier you could always build a module. Also, depending on what you're trying to do, someone may have already solved it so search around at Devot-ee for potential add-ons.
Along the lines of what Wes said about an existing add-on that might do what you need:
Mo' Variables makes a lot of what the Input Class provides available in your templates (ie, GET, POST, GET and POST, and Cookies). It's a very useful add-on.