The first thing you can do is plug in an external hard drive, find the content you really really really can't do without and copy it to the external drive. One less thing to worry about.
Secondly you can look into CCC or alike (carbon copy cloner or something), these are free or paid apps that help you mirror the hard drive bit by bit, giving you a bootable clone.
Thirdly you can just pick specific things out of there I guess. And not make a clone, but just backup certain directories. Maybe those apps support that too.
Finally, you could run a time machine on that usb drive as well. And then on the new machine you can do 'migrate data from time-machine backup' to restore it.
Once you are confident you have a few copies of your data, maybe it's time to wipe the old macbook drive so when you return it your personal data isn't at risk of being leaked to the net.