Try using the default theme again (twenty fourteen or twenty thirteen… either way) and disabling all your plugins… does the media upload still hang once you do that?
If so, you may want to contact your web host….because at that point it’s probably something on the server itself. Unless you’re pretty comfortable with servers, you may be much better off contacting the host and seeing what they say. If you are not aware of it, there is also a WP jobs board here: https://jobs.wordpress.net/ You may be able to find help there.
Do try turning off all the plugins and switching the theme back to default theme first. There’s actually a good chance that will take care of the problem. If uploads work with the default theme and no plugins, you can reactivate your theme, test again, reactivate plugins one at a time, keep testing, etc, and isolate the problem (for that matter, some plugins cause problems that go away after you turn them off and on again… so you may end up turning everything back on and it works perfectly after you do…. definitely give the plugins–>off, theme–>default thing a try before you go digging very far)