打酱油, which literally translates to “buy soy sauce” can be used in two context.
Context 1, which is perhaps the most common one.
It means “it has nothing to do with me, I’m just passing by”.
The term originated from an interview during Edison Chen photo scandal. A reporter was interviewing random citizens on the streets on live television about their opinion on the scandal.
One interviewee replied to the question: “what does it has anything to do with me? I’m just here to buy soy sauce.”
The term soon went viral and has been used to describe an attitude of (slightly annoyed) indifference.
Context 2, it means someone’s child is at the age when they can go buy soy sauce on their own. (helping out with family grocery shopping).
It is usually used in the context to describe someone’s age.
Example: He’s not some kid anymore, (even) his kid (is old enough to) go buy soy sauce (on his own).