I would like to share a more cultural ideology of the meaning.
大 means big.
丈夫 means husband in a more traditional way (think ancient Chinese dynasty). There are other traditional ways to call a husband such as
夫官 (I heard this one from the empress of the founding emperor of Han dynasty in a drama).
A more modern and commonly used word for husband in Chinese is
老公, which can be heard in modern Chinese TV shows.
Anyhow, a common saying in the Chinese language and traditional culture (that still lingers somewhat) is:
A real man sheds blood, not tears.
Or, in Chinese:
男人 大丈夫 流血 不 流泪。
Picture an ancient military commander sustaining sword and spear injuries and still standing strong instead of crying like a sissy. That male then is seen as a "real man".
I suppose, and I don't like this China "big man" attitude, but back then, China was a male-dominated society, so men in those times liked to think of themselves as superior to women and that women were their possessions. It's probably due to this mindset that men like to address themselves as 'big' or 'great' or 'important', hence 'big husband' or
Some arrogant men might call themselves
本大爺 which can loosely be translated to 'big lord me' as in 'Big lord me don't have to pay after eating (at a restaurant)'. Notice how
大 is used to exaggerate one's importance. The equivalent of this in Japanese might be
Men are culturally seen as the owner/master/leader of a family, and they like to be seen as that because a leader is usually recognised as the alpha male, as in a pack of wolves or a pride of lions.
If a man, as the lord/master/leader of a family, cries when he gets hurt a little, he is seen as a sissy or a weakling.
This might be why during the introduction of Han dynasty characters or Hanji/Kanji, Japanese used the phrase:
Are you a big husband? (大丈夫 ですか？)
To mean 'are you okay?' because (as explained above) if you are a big husband, a real man, who doesn't cry when he gets hurt, then you are okay, hence
大丈夫 です or 'I am a big husband therefore I am okay'.
That is my own personal interpretation of
大丈夫 as someone who is Chinese learning Japanese. This phrase clicked for me once I saw the kanji as I was learning Japanese and kept wondering what the heck 'dai jou bu' meant.
One last interesting point I want to say is, the Cantonese pronunciation of
大丈夫 is 'dai jeung fu' which sounds almost the same as 'dai jou bu' (part of also the reason it clicked for me).
Hope that helps.