I'm not referring to the idiom "stare down the barrel of a gun", but rather its parent stock phrase "stare down the barrel of sth".
I found a Q&A form entitled "What does the idiom 'staring down the barrel' mean? How did it originate?" on Quora.com. The answers on this forum do a somewhat decent job of explaining the definition of this stock phrase.
For example, User Raynetta Manees writes:
"When you are staring down the barrel of something, you are faced with the pending danger of it happening. “Having gambled away his entire fortune, John was staring down the barrel of bankruptcy.” The word “danger” in the definition is significant. Proper use of the idiom necessitates that the “something” you are staring down the barrel at is a negative thing, a thing you would not want to happen. So you would not use the idiom this way: “Having met the woman of his dreams, John was staring down the barrel of happiness.”"
My interpretation of this stock phrase's meaning:
So when your staring down the barrel of anything besides the barrel of a gun, it means a potentially dangerous or unpleasant situation is about to happen. Essentially, you have a fast, hard choice to make to deal with the matter at hand (as though mother nature or the urgency of the situation has taken on a tangible physical form to come and threaten you).
I could be wrong, so please correct my understanding if there are any errors.