Custom 404 template file in Drupal 8

You can now use page--404.html.twig. Just be sure you don't set a node as the 404 page. Source: https://www.drupal.org/node/2960810

page--system--404.html.twig (or the equivalent for other 4xx statuses) no longer works in Drupal 8.3 as the 4xx response handling has changed. You'll now need the core patch from https://www.drupal.org/node/2363987 or a similar custom module hook that adds template suggestions for these pages:

/**
 * Implements hook_theme_suggestions_page() to set 40x template suggestions
 */
function MYMODULE_theme_suggestions_page(array $variables) {
  $path_args = explode('/', trim(\Drupal::service('path.current')->getPath(), '/'));
  $suggestions = theme_get_suggestions($path_args, 'page');
  $http_error_suggestions = [
    'system.401' => 'page__401',
    'system.403' => 'page__403',
    'system.404' => 'page__404',
  ];
  $route_name = \Drupal::routeMatch()->getRouteName();
  if (isset($http_error_suggestions[$route_name])) {
    $suggestions[] = $http_error_suggestions[$route_name];
  }

  return $suggestions;
}

EDIT: It's probably nicer to use hook_theme_suggestions_page_alter to modify the suggestions array. See an updated version of this code in https://www.drupal.org/project/fourxx_templates (or https://github.com/ahebrank/fourxx_templates/blob/8.x-1.x/fourxx_templates.module)

The following implementation adds a template suggestion for page, in this case if you create a page--404.html.twig file in your theme, you'll be able to customize the page and works with Drupal 8.5.1

MYTHEME.theme

/**
 * Implements hook_theme_suggestions_HOOK_alter().
 */
function MYTHEME_theme_suggestions_page_alter(&$suggestions, $variables, $hook) {
  /**
   * 404 template suggestion.
   */
  if (!is_null(Drupal::requestStack()->getCurrentRequest()->attributes->get('exception'))) {
    $status_code = Drupal::requestStack()->getCurrentRequest()->attributes->get('exception')->getStatusCode();
    switch ($status_code) {
      case 404: {
        $suggestions[] = 'page__' . (string) $status_code;
        break;
      }
      default:
        break;
    }
  }
}

and create a template called page--404.html.twig and override with your stuff.

OR,
if you want to add suggestions for all error pages, just take out the switch statement.

/**
 * Implements hook_theme_suggestions_HOOK_alter().
 */
function MYTHEME_theme_suggestions_page_alter(&$suggestions, $variables) {
  /**
   * error page template suggestions.
   */
  if (!is_null(Drupal::requestStack()->getCurrentRequest()->attributes->get('exception'))) {
    $status_code = Drupal::requestStack()->getCurrentRequest()->attributes->get('exception')->getStatusCode();
    $suggestions[] = 'page__' . (string) $status_code;
  }
}