So difficult, painful, or unpleasant that you cannot tolerate or endure it anymore.
From Middle English, combining the prefix 'un-' (not) with 'bearable' (able to be borne or endured), which comes from Old English 'beran' (to carry, endure). The word literally means 'not able to be carried or borne.'
The word 'bear' in 'unbearable' has nothing to do with the animal—it's an old meaning of 'bear' that means 'to carry' or 'to endure,' which is why we still say people 'bear witness' or 'bear responsibility.' It shows how the same word can travel through centuries with completely different meanings.
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