Is it possible to carry a 9 foot piece of lumber in an enclosed rectangular trailer that is 4 feet by 8 feet by 4 feet?

Maybe, depending on its exact dimensions. If it's an ideally thin dowel that's 9 feet long and of negligible radius, then it will fit; if it's a beam four feet wide and thick, then no.

First let's see if we can carry it flat:

4^2 + 8^2 = 16 + 64 = 80. This is less than 9^2, so we can't carry it lying flat no matter how thin it is.

However, if we include the third dimension, then 4^2 + 4^2 + 8^2 = 96, and that's greater than 9^2 (81), so it could potentially fit.