Your easiest route is probably to use a plugin like Restricted Site Access (disclaimer: written by my boss) to redirect a user who isn’t logged in elsewhere, like the network home or something else. There are a few other similar plugins out there.
Another option would be to take the code you say you’d put in your theme and make your own little plugin with it instead. Then you can activate it for just that site. You’d need to be careful about which hook you use to check the user and then redirect.