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Looking at the individual kanji according to a dictionary:
丈 means height, stature, length
夫 means husband, man
大 means big, great

They seem unrelated to what these words using the kanji mean:
丈夫 means healthy, strong
大丈夫 means safe, all right, okay

How did these words end up with these meanings?

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up vote 10 down vote accepted

A summary of the gogen-allguide reference for 大丈夫:

丈 refers to a measure of height; about 3m by the 尺貫法{しゃっかんほう} system (traditional Japanese standard) but less in older Chinese measures. One 丈 referred to the height of a man.

夫 here means 'man', so when these words were imported from China, 丈夫 meant a fully grown man and 大丈夫 was a great/splendid man. From there 大丈夫 started to be used to mean "exceptionally strong", "exceptionally reliable", "exceptionally healthy", and from there the other meanings developed.

(大丈夫/丈夫 do not mean the same things in Chinese).

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3  
In modern Chinese, "大丈夫" means "big husband". – Joe Z. Jan 29 '13 at 4:08
    
@JoeZ. Not quite. It means "manly man" in Chinese, where Japanese would say 偉丈夫 instead. – broccoli forest Jan 27 at 20:09

I think they seem related.

A tall man is a strong/healthy man. A really tall man is someone who is well/alright (i.e. without any problems).

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I would like to share a more cultural ideology of the meaning.

In Chinese means big.

丈夫 mean husband in a more traditional way (think ancient Chinese dynasty). There are other traditional ways to call a husband such as 相公 and 夫官 (I heard this one from the empress of the founding emperor of Han dynasty in a drama :D)

A more modern word for husband in Chinese commonly used is 老公 which is what you hear in modern Chinese TV shows.

Anyhow, in the Chinese language and traditional culture but still lingering somewhat, a common saying is:

A real man shed 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 stand 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 is a male dominated society, so men in those times like to think of themselves as superior than women and that women are their object of possession that they own. It's probably due to this mindset men like to address themselves as 'big' or 'great' or 'important', hence 'big husband' or 大丈夫.

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 is used to exaggerate one's importance. The equivalent of this in Japanese might be 俺様 (ore sama)

Men are cultural 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 like in a pack of wolves or a pride of lions.

If a man, being 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 get's 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 :D

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.

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