On the supported mac download Sierra, and before it starts the installation look in the Applications folder for Install macOSSierra.app and copy it to a USB stick, or send via network to your old iMac.
You will need a copy of Sierra before you start, as your iMac is too old it won't let you download it so you will have to use another mac that is supported to do this. Please make a backup of your current system prior to attempting this. So if you look at this, 977128/ and follow the instructions from dosdude1 and foxlet you should be able to get Sierra running on your mac.
