I have been in OC since the content has been out, and I open up chests pretty frequently. I have gotten 4 naturally, and got a 5th one off the market board (someone messed up their offer).
You're better off just selling the carrot to a viera player who wants the achievement. There are other achievements in the game that are long term efforts, but none of them require both a time commitment and literal billions of gil to do. Doing it "naturally", even accounting for the fact this will probably also be done in North Horn, will take you thousands of hours at the drop rate. (And you have to actually do the content to respawn chests) And the rewards in the chest are most of the time not even worth it.
