We live on MUCH less then that per week, as a family of three. I know that is a big difference from a family of 5!! We do own a home, and our mortgage is less then your rent. We don't have debt or student loans, so that is how we do it. We own one car and are on the fast track to pay mine off in 3 months. We bought my car as gently used, for the same reason you did. It was a great deal as far as payments and insurance go, and we need a reliable family care that will last many years.
We are slowly building a savings. Nothing more then 15-20 a week baseline, but with little extras, it adds up quickly. I'll show you some practical ways we put extra money in savings. We ONLY budget for my husband's 40 hours a week. If he get any overtime, we do not spend that on other things. It goes straight into savings. If he gets a small bonus (not common) it goes into savings. If we get a tax return, straight into savings. I have a strict budget. My husband works a few blocks from home, so we don't spend much in gas. If we spend less then what I budgeted in gas, straight into savings. We buy big bags of fairly expensive dog food. (Our dog requires specific food, she has a stomach issue.) We got a coupon in the mail, for a free bag. That $40 we would have spent on dog food without the coupon, straight into savings. I have a budget for $200 (yes, we eat very healthy) monthly groceries. I try very hard to keep it at $150. When we get to the end of the month, if I've only spent $150...the extra 50 goes into savings. We have a VERY strict Christmas budget. I try to stay far under the budget. If I stay under, the extra goes into savings. When we moved into our home we made more money and signed up for cable, the first time we've had it. We are locked in a contract now and make less money, but our contract ends in January. When it ends, we still will put it into the budget, and out that extra into savings. Whatever we don't spend from our budget, goes into savings. I find myself often transferring as low as $2 into savings. Whenever we have a little left over from the budget, we do that. Let me tell you, those little amounts add up very frequently. I am not above transferring .50 into savings, if the extra is there. I have a checklist of things we are saving for. My son's future activities are on there. (He is 2 1/2.) Things like soccer, karate, college, etc. The college one will be on the checklist until he is there, but the others gets checked off...slowly, but surely.
Could you get a job outside of the home? I have never made much money inside the home, it's just not usually very possible. If my husband ever get too tight in the future, I will take on a weekend job. Could you do that? Are your children too young, for you to work outside the home during the week? What do you mean you won't make much for one child?!!! If you watched one child even 4 hours per day, and only charged $5.00 (you could charge more then that) an hour, you would make $400 extra dollars per month!! Imagine if you watched a child a FULL day and charged more then that!