Rails Polymorphic conundrum

I'm making a database that requires many-to-many associations. Here's the design:

There is a class called 'Owner' that owns many 'Boxes', 'Things', 'Doodads', and 'Thingamabobs'

'Boxes' can have many 'Things', 'Doodads', and 'Thingamabobs', and each 'Thing', 'Doodad', or 'Thingamabob' can belong to many :boxes. Do I just need to make three separate join tables, or is there any way to create this with a polymorphic association like so:

BoxesContents   box_id   content_type   content_id

(content being a :thing, :doodad, or :thingamabob

Is it possible to do this without the use of has_and_belongs_to_many?