"the bear" would mean that a reference to the hapless bear has already been established, and that would require reordering of the czech sentence, preferably along with adding the demo pronoun as well: "zabil (toho) medvěda holýma rukama." try searching for the contextual differences associated with the (in)definitiveness of the bear in "he killed a/the bear with a knife" on google books.
I would suggest in this sentence the emphasis is on killing a bear without tools such as a knife. It does not mean he had nothing on his hands he could be wearing gloves. In this type of sentence though 'his' would usually be used. Although when it is not so dramatic you could say something like she washed up with bare hands meaning she didn't wear gloves. I do not think there can be any question as to whose bare hands are being used. I am an English English native speaker
Just when I thought that instrumental declensions seem easier than the other declensions, Duo presented me with “rukama” which didn't seem to fit. So I looked it up ( https://cs.wiktionary.org/wiki/ruka ) and discovered that “ruka” declined in singular and plural as well at dual!!!!! What kind of words decline in dual? Only body parts? Socks? Windshield wipers? Do the adjectives for "dual" nouns decline in the same way?