It also depends on whether one also has a basic store of things like oil, flour, herbs and spices etc. And to do it very cheaply, you need to budget over a month rather than weekly so remains of large packs of pasta, rice etc can be carried forward to next week. Just supermarket shopping, I could eat very well on £20 per week if I had to buy in basics as well, probably about £15 or a little less if I had store cupboard items in.
When I was dirt poor, I collected veg that fell off market stalls on to the floor, and cooked them in the pressure cooker with free dog bones from the local butcher. If you live in the country, road-kill pheasants are a free source of protein 