How do I pass an object from javascript to PHP? I set up a hook_menu like this:

$items['fleet/construct'] = array(
  'page callback' =&gt; 'construct_fleet',
  'type' =&gt; MENU_CALLBACK,
  'access arguments' =&gt; array('access content'),
);


Then I set up a js file with this:

$.post({
  url: '/fleet/construct/',
  data: {new_ships: [1,2,3,4], destroyed_ships: [5,6,7,8]},
  success: finalizeFleetResponse
});


How do I access the data from inside my menu callback function?
You are using a POST, it is the same way you, use 

$new_ships = $_POST['new_ships'];
$destroyed_ships = $_POST['destroyed_ships'];


Read more on php POST.