Not true. The secret is a well stocked pantry. We have never had a bare pantry, so keeping it updated only means adding a fresh spice every few months or a few tins of whatever a week. We easily have 20+ spices at any given time, different grains, loads of different pastas. We make our own sauces and spice mixes, everything except breads really is cooked from scratch. We meal plan accordingly and use loads of fresh vegetables and herbs. We make most of our cleaning products (more for the environmental impact than cost), I make a large amount of my own beauty products too. In summer we cloth (honestly can't keep it up in winter). It's very possible for a 4 person family to get by on $100-$150 a week, but it's a lot more effort. Worth it though IMO.
I agree with the well stocked pantry - but to get down to 100 average (I wish I could find where I read is achievable), if you have nothing in your pantry to start with I think it takes alot of months before you would get near $100 a week average. Some weeks we definitely are as low as $100 if not lower - but have now been recording for 2+ years (my figures are an average across that time) and now have a reasonably stocked pantry but to get to an average of $100 we would need quite a few months of only spending $80-$90 but then pantry items run out - personally I think an average of $100 a week starting from empty cupboards for a family of four (especially once you get to the point where both children eat a decent amount) would be pretty hard to achieve with current food prices. An average of $130, 140 is probably more reasonable.

(Most sites with info about food budgeting do all talk alot about the well stocked pantry, but not all seem to take into account the cost of the items in that pantry to start with, but they are not free they should be taken into account when budgeting)