The “Four Olds” were anything related to the old China before the Cultural Revolution.
The Cultural Revolution in China resulted in the elimination of anything and anyone that had power before. Literally, the Four Olds were “Old Customs, Old Culture, Old Habits, and Old Ideas” and the goal of the Cultural Revolution was to ride the country of them (Wikipedia). Chairman Mao established the Cultural Revolution when Ji-li was young, where she heard about the “evil influence” of the “Four Olds.”
Chairman Mao told us we would never succeed in building a strong socialist country until we destroyed the “Four Olds” and established the “Four News.” (ch 2, p. 21)
Of course, this also meant not educating the youth. It meant destroying storefronts that were part of the old order. It meant families giving up their heirlooms. It was a very frightening time for Ji-li.
The "Four Olds" refer to old ideas, old cultures, old customs and old habits of the capitalist class. Any objects or ideas related to the old, feudal China, as portrayed in the notion of the "Four Olds", were ruthlessly destroyed during the Cultural Revolution.