No, it is not possible. To prove this conclusion, let's color a $6\times 6\times 6$ cubic in the following way.

Then, there are a total of $8\times 14=112$ dark unit blocks and $8\times 13=104$ white ones. Meanwhile, each $1\times 2\times 4$ block will cover an equal number of dark unit blocks and white ones. Hence, the total number of dark unit blocks must equal that of the white ones in order for a cube to be constructed. Therefore, we conclude this is an impossible task.