I spent a minute thinking about this, answer might not be right, but this is definitely the right methodology.
1 barrel of oil = 31.5 US gallons
1 gallon = 3.7854 liters (standard conversion)
1 liter = 1000 centimeters cubed (by definition)
SO 1 gallon = 3785.4 centimeters cubed.
SINCE 1 centimeter cubed = 10^7 nanometers cubed,
1 gallon = 3785.4*10^7 nanometers cubed = 3.7854*10^10 nanometers cubed.
Since the oil settles over the ocean at 100 nm thick, think about the entire surface being covered by 100 nm^3 blocks of oil.
This means that instead of a surface covered by 3.7854*10^10 one-nanometer blocks, (which would be only one nanometer thick), you'll get a surface covered by 3.7854*10^8 blocks that are one nanometer on each side and 100 nanometers thick. Does that make any sense? It would be easier if I could draw it.
This means that the surface is covered by an area 3.7854*10^8 nanometers by one nanometer (imagine all those blocks stretched out in a huge line).
This has a surface area of 3.7854*10^8 nm squared per gallon. Since 10^7 nm = 1 cm, 10^7 nm^2 = 1 cm^2, and 3.7854*10^8 nm^2 = 37.854 cm^2.
Thus, one gallon of oil results in 37.854 cm^2 area covered.
31.5 gallons of oil gives you 1192.4 cm^2 area covered.
since 100 cm = 1 m, this is 11.924 m^2 area covered.
since 1000 m = 1 km, this is .011924 km^2 area covered.
This, off the top, seems too small to me. I think there is a mistake in there somewhere (I did this by hand on a piece of scrap). But this will lead to the right answer. Hope this helps.
By the way, killer problem. Hope that's extra credit
I know there are some math whizzes on the boards. Good luck! 
