This is might be the coolest DIY project for a college student. Milk crates are the same size anywhere you go, so this should work in all parts of the country.
As you already know, a queen-sized mattress is 80 inches long and 60 inches wide. A standard milk crate is 13 by 13 inches, with the square side 11 inches and the rectangle side at 19 inches in length.
If you calculate the dimensions, it would come to five rows of eight crates per column. You would need 40 crates to build a queen size bed frame.