Is that 4 barrel halves or is it 4 barrels cut in half to make 8 half barrels?
Are the 55 gallon barrels?
Will this system have a sump tank or will it just be grow beds above fish tank and thus why you need the tank to be shallow?
If 4 barrel halves (that is two 55 gallon barrels cut in half, about 100 gallons of grow bed) and assuming the towers are just trickle through, then I would say the 100 or 150 gallon stock tank (be sure not to use a metal stock tank) would be fine as long as you only stock it with 10-15 fish intended to grow out to say 1 lb each for your first season.
If you are talking 4 barrels cut in half, or 200 gallons of grow bed then I would probably say the 300 gallon rubbermaid stock tank would be the way to go (and this will give more temperature stability than the above mentioned tank, heck you could use the 300 gallon stock tank with the first set up but you still only get to stock it with the amount of fish your filtration can support.) For 200 gallons of grow bed plus the towers I would say stock 25-30 fish the first season (again figuring on a 1 lb grow out type of fish.)
I'm planning on about as many grow beds. Currently, I'm working with two of these Tractor supply steel stock tanks (advertised volume: 169gal, 6' long, 2; wide 2' high), and an additional wood/pondliner construction sump/crayfish tank. because they are galvanized they need to be coated, but they were ideal for the space I had to work with. Planning on a total of 8-10 half barrel grow beds....