I suppose you mean 'while they are hibernating', as they normally breathe through their mouth and nose. In fact they are renowned for their sense of smell. Some of the Native American tribes have developed these words of wisdom: If a tree falls in the forest, the hawk will see it, and the deer will hear it, but the bear- the bear will smell it...
Hibernation dens are not totally sealed off - many's the explorer caught in bad weather who found a great place to hide from the weather, only to find it occupied by a slumbering bear. Some bears do not go into total sleep, they get reallly lethargic (slow) during the times of colder weather.
So the answer is... through their nose and mouth.
From the lighter side: Unless you mean the Baylor Lady Bears Basketball team. They get air by practicing those tedious jump-squats on a regular basis.
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