How can I temporarily remove content from all pages, but leave up URL?

Depends on what you mean by “content”. If you just want to remove the post body then…

function remove_body_wpse_97291($content) {
  return "Content temporarily removed";
}
add_filter('the_content','remove_body_wpse_97291',100);

If you want more aggressive content removal…

function remove_all_content_wpse_97291($content) {
  include('license.txt'); // example loads the WordPress license agreement as raw text; not pretty
  exit;
}
add_filter('template_redirect','remove_all_content_wpse_97291',1);

The URLs stay the same but you replace content.

You could also redirect with (maybe) a 307 Status code, or 302 possibly,…

function redirect_all_content_wpse_97291($content) {
  wp_safe_redirect('license.txt',307);
  exit;
}
add_filter('template_redirect','redirect_all_content_wpse_97291',1);