Ummm...yes. You can. Our family's budget is about $450-550/mo for our family of four which is two adults and two teenage boys. That amount includes non-food items like TP, laundry and dish soap, shampoo, toothpaste, etc. so we are well below the $4/day ($120-124/mo) per person limit you're talking about.
The problem with this video is she bought a head of lettuce and a bottle of salad dressing which put her over her $4 budget, but the lettuce will make 6-8 servings of salad and the dressing has about 20 servings, so these will last much more than one day.
Also, buying fruit in that tiny little cup is the least economical way to buy fruit. Ever. Ditto the bagel--buying a bagel is also the least economical way of buying bread.
I am adamant that we eat 2-3 fruits and/or vegetables at EVERY meal. I am equally adamant about using whole-grains and non-processed or ready-made food. We eat meat 3-4 days a week and our protein for the other 3-4 days a week comes from eggs, dairy and legumes. Preparing food and eating meals are important daily events in our family, with everyone participating. We are well-fed--physically, spiritually and emotionally because we are conscious of and deliberate about our food and food choices.
Eating on a budget requires that one stay away from ready-made and processed foods, cook from scratch and shop sales. It requires a re-education or re-orientation of our priorities.
So, yes, not only can you eat on $4/day, you can eat well and healthy. The hunger in our nation is not a shortage of food or money, but of consciousness, morality and knowledge.