Is it possible to have an entity with fields, which doesn't use bundles?
Technically, yes! For each entity type you must have a bundle. 
Since bundle is the implementation of entity type. If you define your entity type without a bundle the bundle will be of the same name with entity types. For example user entity doesn't have a separate bundle so it uses the default one with the same name of entity type. 

Please go through this and this for details.