Is it possible for humans to digest and live on food that isn't based on DNA or RNA?
This might be flagged for being somewhat of a subjective question, but to survey human diet:

Mnemonic SPONCH can be used to sum up the necessary building blocks of life, in the elements Sulfur, Phosphorus, Oxygen, Nitrogen, Carbon, Hydrogen. There are others, trace elements, which are necessary and come from diet (selenium, for example). These elements are used in building our biological macromolecules: Carbohydrates, lipids, nucleic acids, and proteins. The molecular interactions between these macromolecules, their cofactors, etc. are essential for life. Some of these compounds, however, can't be made in the human body from scratch; hence essential amino acids, and the necessity to consume foods which provide metals, carbohydrates, fats/triglycerides, etc.

So, at this point our body is optimized to incorporate terrestrial foods. Sometimes, though, we can see that the body will sometimes accept analogs to normal compounds, although this is often to disastrous effect (like chemotherapy in cancer). The real question, then, would be: does said extraterrestrial food compose of necessary biological elements or analogs in a way our body can actually use this food for sustenance? That, however, is 100% speculation as we have no reference in even a single case to base our assumptions.