Is it possible to create a Django app which only reads data from external database and provides REST API?
There is no technical difference between external and the own database as pointed out by Kalus D. any database which is online and the username and password of it is known (means provided in django's setting.py) is Django's database. There can be multiple Databases of this type. so that's it.
But I don't think you mean the above. I think What you mean is collect data from external APIs or web pages e.g Wikipedia Rest API. then process that data like filter it, combine different data or anything you like and then send it to the user.
then yes it is possible but you don't need DRF for it. DRF is mainly applicable when you want to serialize your won models data. you can just send response in JSON format insted of html without using DRF.
In other words - I want to create a simple Django Rest Framework app which preferable doesn't need it's own database.
This Django API app should only provide data for some external frontend app (e.g. Angular/Webpack stack).