LORELAI: Call me when you get home, and please be careful.
RORY: I will.
LORELAI: I mean it Timmy, no falling down the well.
Lorelai is referencing an old joke relating to the television show Lassie, earlier discussed.
In the show, Lassie would bark to give warning of danger, with her human friends apparently understanding exactly what she was saying. Thus it was parodied as, “Woof, woof!”, “What’s that, Lassie? Timmy’s fallen down the well?”. The joke relates to the 1957-1964 period, when the little boy on the show was Timmy Martin, played by Jon Provost (who called his memoirs Timmy’s in the Well: The Jon Provost Story).
In actuality, Timmy never fell down a well, although he suffered a number of similar situations, such as falling in a lake and getting trapped in an old mine, a pipe, and down a badger hole. The list of Timmy’s perils is very long, and includes wandering onto a minefield and being exposed to radiation, not to mention more mundane concerns like tigers and bears. Lassie did once get stuck down a well herself, though.