Given $sid is your subqueue id:
$entity_subqueue = \Drupal::entityManager()->getStorage('entity_subqueue')->load($sid);
Loads the subqueue object. You can manipulate this to retrieve or update your entityqueue.
To get the items:
$items = $entity_subqueue->get('items')->getValue();
To update the items, add another item to the $items array and then save the entity object:
$items = ['target_id' => 69];
You can check in the database that your entity queue is updated:
select * from entity_subqueue__items where bundle = :sid;
I would like to also add that there is a great alternative to using entityqueue in Drupal 8: the Config Pages module. It allows you to create entity bundles that have only one entity. You can then add any field type you like, including entity reference fields. I recently replaced an entity queue on a project with a config page with an entity reference field, and I preferred the user experience.