With smart shopping, I'd imagine it's possible to low carb on that type of budget, especially only feeding two people. You really don't need things like the expensive low carb goodies or even cheese, really. Just fresh meats and veggies. You can often get frozen or canned veggies on sale.
If you get lots of chicken (whole is cheaper than already cut up), and cheaper cuts of meat or buy meats on sale, you can really stretch that grocery dollar. Eggs are a protein bargain; one of the local groceries this week was selling extra large eggs for 59 cents a dozen! Use your crock pot on the cheaper cuts of meat...they come out juicy and tender that way. Take advantage of meat sales and stock up if you have a freezer.
So...if you're willing to do a simple version of low carb, it's possible to do it on a budget.