How can i save a block_field field programmatically?

7 years later! In case you need to do it, heres what's in their test code :

// Create a referencing entity.
$referencing_entity = $this->container->get('entity_type.manager')
->getStorage('entity_test')
->create(['name' => $this->randomMachineName()]);

$referencing_entity->field_test->plugin_id = 'block_field_test_access';
$referencing_entity->field_test->settings = [
'label' => 'Custom access.',
'label_display' => TRUE,
'access' => TRUE,
];
$referencing_entity->save();