Associating ActiveRecord and non-ActiveRecord objects

Is it possible to define a model that is not backed by a database
table and still relate other models to it? Will relations between
instances of these models persist?

For instance, say I have a Cat model that does not inherit from
ActiveRecord and an Owner model that does. Can I assign a particular
cat to an owner? The next time I load the app will that pair be
remembered?

No.

Unless you roll your own solution or persist the cat at the database
in a way or another.