My wife and I were on a very strict budget so we often skipped many of the places to eat right outside of the Vatican walls. Many catered to tourists which usually means high prices and low quality food. Any place that says 'We accept credit cards' or 'Engish spoken' is probably a waste of your money.
Vatican City itself offers no dining options.
My wife and I found the market on Via Andrea Doria prior to visiting the Vatican Museum. It was an open air market and had plenty of fresh fruits, vegetables and some meats. We grabbed some bread and sandwich options and a carton of juice from the nearby Meta supermarket for our lunch that day. We interacted with some locals and spent about $5 each. We ate it while looking at San Pietro from the square. Couldn't get that view from any restaurant. =)