Is it possible? How hard would it be to do so?
Yes, it is possible. It is not very hard but it requires a good comprehension of boot sequence.
The "copy" sequence is something like:
The last point is the hard one. You are copying a full-fledged linux system on the stick so you can't use vfat partitions. If you use ext2 partitions, you'll can use syslinux otherwise you'll have to use grub.
In any case, the result will be very slow because normal distributions are not optimized for this kind of use. If you do not need a temporary solution, try some live distribution. For example, it is very simple to create a customized system rescue cd usb-stick adding whatever software you need on it.
A little hint: use