Hplip cannot be installed on system with disabled root-user (password incorrect, can't install dependencies)

This thread is derived from an askubuntu-thread There the thread-opener tried to install hplip-libraries, but failed because hplip handled his system as a pure Debian one, instead of an Ubuntu one. A very related problem affects users of distros other than Ubuntu -- because Ask Ubuntu is for Ubuntu only, I am opening this thread here to hopefully help some people of other distros. The symptom of the problem is: during the execution of hplip-*.run, you are asked for your root password, but, entering your correct password, the installer will tell you your password is wrong. This only affects people with a disabled root account (people using sudo). How can one fix this problem and install the hplip on a machine that has the superuser disabled?