I think this would be doable, just rather boring and unvaried in the dietary department. I can get a pound of pasta for $1, add in pasta sauce for about $1/pound, and we're at $24 for the 2 pounds/day of food that the average person eats. That leaves enough for a few bell peppers or onions to try to avoid scurvy. There's no tax on most food where I live, and H2O works for beverages, so it's a go.
In college, I actually did live on such a diet more than I probably should have, though more so because it was what I had reliably in the house than due to budgetary reasons. As such, I've somewhat burned out on it, though tonight I did have pasta, tomato sauce, green pepper, and mushrooms (with a few meatballs) for dinner.
Rolled oats might be a good option. I've heard dry cereal is among the most expensive breakfast foods by weight, so I'd likely skip that if trying to minimize the budget. I do often add cinnamon, brown sugar, or both to rolled oats when making them myself. I've also found that Bob's Red Mill's recommended portion is too large for the bowls I have, so I have to make half-portions to keep it microwaveable.
Potatoes can also be pretty cheap. I've occasionally eaten plain baked potatoes at lunch when I forgot to bring condiments along. Not exactly thrilling, but economical. And I can get frozen vegetables for $1/pound pretty reliably - soybeans, green beans, corn, a pretty good variety. Protein is a noticeable problem, but a vegetarian might be able to do a better job of that than I would.
Granted, I usually spend more than $29/week on food, even when I cook for myself all the time, and it's not easy to get a balanced diet on that. The cost of living is fairly low here, which likely helps. But what you're used to also makes a difference. Gweneth Paltrow probably spends more on food during the average week than I do, and may well do less home cooking as well. So, I'm not surprised that someone who's used to earning $1 million/year would struggle with this adjustment. Arguably, it's too low to begin with, too, particularly in expensive neighborhoods. In more reasonably-priced ones? Probably doable, but with less than ideal variety and nutrients. And if you're really poor, it's more likely that you'll live in a food desert with no reasonably-priced grocery stores to boot. Far from an ideal situation.