There are object-oriented database engines available. Cache is one of them, for example. They have the OO properties that you are looking for. If you absolutely have to have polymorphism in a database, go with an OODBMS (Bing/Google that term if you don't know it), not with an RDBMS.

It is theoretically possible to do inheritence and polymorphism in an RDBMS. But you can't attach methods to an object, and columns aren't really properties. OODBMSs allow you to do proper inheritence, polymophism, properties, and methods. That's what they're for. They tend to be less effective at the things RDBMSs are good at, like data analysis and reporting, data mining, set-based queries, and so on. But they are good for OO data storage.