Make sure you navigate to a folder within your dropbox where you want to do stuff, for example copy files out.
Now, click in the addressbar on an empty space. This should allow you to enter text in it manually and copy the address.
Clear the addressfield and type the following command in its place:
subst j: .
You may change j: to any driveletter that is not in use on your system.
After completing this command, a new drive forms. If you used j:, then the new driveletter is j:.
Go to that driveletter. You can now do anything you want because the path is no longer c:\Users\YourUsername\Dropbox... but j:...
So you don't really need to shorten the folders if its just for a backup you want to make.
The deeper you go into the folder structure, the more chance you won't run in a "path too long" kind of error. You can also repeat the process many times, changing the driveletters to unused ones.
Want to get rid of the drives, just reboot your pc. The drives are only there for the duration of that windows session. Alternatively, open a command prompt and type
subst /d j: and substitude j: for the driveletter you used. If you created 3 drives, run the command 3 times, once for every driveletter.