Why do we say "What do you do for a living"?

What does "a" (the indefinite article) imply in this context?

Solution 1:

In that context, "a living" refers to the subject's profession/job. So, that question is asking:

What do you do for your profession?

You can also say things like:

I build houses for a living. (Your profession is building houses)

I found this definition for living:

an income sufficient to live on or the means of earning it.

Source

The indefinite article is simply referring to the fact that there are multiple things that a person can do for their job. A definite article would imply that there is only a single thing a person can do to make a living.

Solution 2:

To my knowledge, it is a shortening of the phrase "a living wage."

People will also use this expression to say things like, "It's a living."

But, you can also say "I make my living . . ."

The short answer to your question: Living in this case is a noun. And, therefore needs the article to designate it as such, instead of being a verb.

If you asked "What do you do for living?" (Gerund of the verb) The answer would be akin to "Breath, eat, drink, etc."