How can I get $_POST data in a callback?

They access $_POST in the same way other functions do.

Take file_ajax_upload() as example; it contains the following code.

  if (empty($_POST['form_build_id']) || $form_build_id != $_POST['form_build_id']) {
    // Invalid request.
    drupal_set_message(t('An unrecoverable error occurred. The uploaded file likely exceeded the maximum file size (@size) that this server supports.', array('@size' => format_size(file_upload_max_size()))), 'error');
    $commands = array();
    $commands[] = ajax_command_replace(NULL, theme('status_messages'));
    return array('#type' => 'ajax', '#commands' => $commands);
  }

The function is a page callback defined in file_menu().

  $items['file/ajax'] = array(
    'page callback' => 'file_ajax_upload', 
    'delivery callback' => 'ajax_deliver', 
    'access arguments' => array('access content'), 
    'theme callback' => 'ajax_base_page_theme', 
    'type' => MENU_CALLBACK,
  );