Translation:The whole family listened to the radio.
English wants the definite radio while Swedish wants radio to be indefinite for the activity of listening to the radio. If you say "på radion" it means they were listening to a specific one radio receiver.
Consider this: in english we say we watched tv, not necessarily the tv. The subtle differences in the two languages.
Why doesn't "hela familjen" take an article? For example, wouldn't one write "Den stor familjen?"
Some adjectives are used without an article, one of them is hel. By the way, your second example should be den stora familjen with the adjective in the definite form as well.