If I'm running a version of Ubuntu more recent than that on which elementary OS is based, can I install elementary's components onto it for a working elementary OS setup?
You could install the elementary desktop and all the standard applications on top of your existing Unity desktop with these commands:
sudo add-apt-repository ppa:elementary-os/stable
sudo add-apt-repository ppa:elementary-os/os-patches
sudo apt-get update
sudo apt-get upgrade
sudo apt-get install elementary-desktop
You'll probably have to logout and switch to Pantheon in your display manager.