I hate being one of the first to respond to these things because the first few folks always get torn apart but here goes. I'm going to ignore the LA option.

First some math to consider:
Las Vegas to Chicago is 1750 miles (according to Mapquest)
Las Vegas to New Orleans is 1740 miles (same source)

Assuming an average gas price of $2.50/gallon (locally for me it's $2.35 now)
- If the car gets 30 MPG you will spend $145.83 in gas
- If the car gets 25 MPG you will spend $175.00 in gas
- If the car gets 20 MPG you will spend $218.75 in gas
- If the car gets anything less than 18 MPG you do not have enough money for gas to Chicago or New Orleans much less food, water or anything else.
AND this is all assuming gas actually does average $2.50 which is probably not realistic because if there is an “event” gas prices will go up quickly AKA Katrina.

Possible alternatives to car travel? – None I can see or would be willing to do.
With only $250 and no credit card flights out are beyond the budget even if they are flying. And according to my quick check of Greyhound.com a bus would cost $242 ($151 for the adult and $91 for the minor child) and Amtrak quotes a train to Chicago at $409. Therefore both the bus and train are also beyond the budget.

Therefore I will assume the car gets 20 MPG so I am spending:
$218.75 Gas
$ 4.98 24 Pack of 16.9oz Bottled Water
$ 2.98 Box of 10 Breakfast Cereal Bars
$ 2.98 Box of Peanut Butter
$ 2.50 Crystal Light Hydration Drink Packets (to keep electrolytes up)
$ 2.99 Box of 24 Stay Awake caffeine pills

Total Spent: $235.18 leaving me $14.82 in cash.

The above ASSUMES I have my EDC, PSK, and FAK with me…if not for some
unknown reason I did not have ANY of these three things then I would:
Eliminate the Breakfast Cereal Bars and buy:
$5.00 pre-paid phone card
$5.00 2 Emergency Space blankets
$3.00 Bottle of Advil

Ok now I’m out there on the limb, let the criticism begin…