I never liked that joke. Obviously, as an individual, you can't eat an elephant in small bites, as (assuming it's dead) it will start spoiling very soon after you started biting into it (and if it's still alive, you might have an even bigger problem).

So the actual solution would probably involve a lot of logistical planning and preparation on how you would carve it efficiently and then refrigerate and/or distribute all the pieces.

Not doing so, and only thinking one bite at a time, would be a massive waste of the elephant.