So, I know this is question was asked some time ago, but, I have another theory all together, and kinda goes along with user17976's answer. If, let's say, Ross first asked Carol out around 1984-85, he's saying it's been a while. So, this may be a little far fetched, but I think it may be a reference to an old TV show around this time frame, called "Misfits of Science". In the second episode "Your place or Mayan", toward the end, one of characters on the show is trying to save one of the others. The one who is being saved yells out "Billy, don't be a hero", as he swings on a rope trying to save her.
Why, you ask, do I believe this may be what Ross is referring to? For one, Ross is considered a science geek, so it's possible he would have been interested in that particular show. And second, that show, "Misfits of Science", starred an up and coming actress by the name of -- Countney Cox. Perhaps the line was thrown in as a nod to how she got her start on a cheesy (and it was very cheesy) Sci-Fi show.
As a said, a little far fetched, but it's entirely plausible. You can check the episode out on YouTube.