I think the answer is just: No
Everybody is welcome to correct me if I'm wrong.
No, as far as I know this is not possible at the moment.
You can, however, download a database backup and restore it on a local Postgres DB.
I agree with above answers, you cannot directly access the database.
I am not sure why you want to do this but if there is some specific requirement then not sure if you have thought of either publishing an API endpoint or an OData service which can expose some required entities or all entities in the database?