Wow. Our food prices are nothing like that, and I'm curious how much is regional difference and how much is ineffective shopping. If we pay more than $2/lb for conventional meat, it's a splurge.
If you haven't already, cut back in other areas to make room for food. Wearing secondhand clothing won't have a long-term effect on your health. Cutting cable or netflix is likely to have a positive long-term effect on your health. But I know very well that this is easier said than done for people already on a limited budget. A lot of "budget" paleo advice seems to be written by people who may have to watch their spending a bit in order to make payments on their new car and 4000 sq ft house, not people who are struggling to make the bare minimum - for whom immediate survival nutrition has to take priority over optimal future health, but who are still trying to make the best choices possible within that constraint.
Fully investigate your shopping options. Try neighboring (or even not-so-close) communities - you may still have to do some fresh-food shopping at local stores, but if you can do large freezer-stocking trips occasionally, it could be worth the gas money. Try stores you don't normally shop at. See if you can find grocery outlet, ethnic food, or restaurant supply stores. Find someone with a membership to Costco or other membership food store and see if you can go with them occasionally. Search craigslist or do a bit of driving of country roads and see if you can find stuff - we get eggs for $2 a dozen from a local farm - not free range, but raised outdoors, and much higher quality than conventional store eggs at not all that much higher a cost per serving. (We still get conventional store eggs too.)
Shop the sales, and look for things that have been priced to sell quickly. Look at the grocery circulars that come in the mail and at internet coupon sites - that can help you know which stores to go to without having to drive around. Most of the coupons are for processed junk, but occasionally there will be something useful - we recently got free eggs and sausage, and someone mentioned a $1 off produce coupon from Safeway.
If there is something you eat a lot of, ask around at various stores about bulk/case discounts (coconut milk, shredded coconut, canned fish, and other preserved-but-ok things like that are prime candidates for this). Compare with online prices.
Often stores will discount meat that is near its sell-by date or produce that is on its way out - get that, and then either use it or freeze/preserve it immediately (Our meat is generally 50% off, and we recently got cabbage for $0.25/lb and made sauerkraut). Be adventurous with cuts of meat you don't know how to cook, cheap veggies you're not familiar with, and so on.
Look on the internet for things that aren't available (or not available at a reasonable price) locally. Make sure the shipping cost doesn't outweigh the price difference (preferably order from somewhere with free shipping).
Limit food waste. Don't let stuff go bad - freeze it if you're not going to use it quickly. Make stock from the bones from any meat you get, along with veggie bits that you don't use.
Look into discount/free food programs (often run through churches) if you're really struggling. You may have to look around a bit to find out which actually has food you can eat - some have produce and stuff, while others are likely to be primarily processed food.