From what I know it would take quite a bit of overriding to get it to work, as joomla does not work "ajaxified" by default. I would look in the joomla extensions directory to see if there is anything like this already done. a template can possibly do this by default (I have seen something similar in wordpress)
http://extensions.joomla.org/search?q=ajax
There is a number of plugins that add ajax functionality, though how closely they match what your goal is can be another matter.
To code this in yourself, you would need knowledge in jQuery/JavaScript and create the template to override links into calling pages though ajax and overriding joomla's functionality to stop it from including headers/footers in all extra pages (just the html you need).
Going further in depth would take quite some time. I recommend looking for a plugin that might do this by default, this way you won't have to spend so much time changing how joomla loads pages.