I would like to install Ubuntu (or Lubuntu, more precisely) on a machine where BIOS does not allow be to boot from an USB (and there is no CD/DVD). I have browsed the alternatives in here, but they are all quite advanced, or at least far beyond my understanding of Ubuntu. Is there any relatively simple way to make run an .iso off a small partition of the harddrive, or loading in into the RAM somehow, to make it work as like if I had run it from a USB startup disk? The computer currently has Ubuntu 14.10 installed, but I'd like to all completely fresh install, wiping everything out.
Absolutely you can run from a hard drive! A hard drive is just an internal version of a USB drive (for your purposes) so you can do this no problem. Just throw the iso onto a partition and you should be good to go.
NOTE: I haven't tried this myself but I know of people who have.