Is it possible to make polyethylene in a high school laboratory?
Not polyethylene, but I recall doing an experiment in school where hexamethylenediamine and adipic acid were poured carefully so one was layered on top of the other.  They did not mix in bulk, but reacted at the contact surface to form a Nylon-type polymer barrier.  We poked a rod in to pull the polymer out (enabling more reaction) and wind it around the rod!