How do I take a full image backup of an Android device?
Yes, you can, by taking a full NANDroid backup. This takes a backup of everything, including the app data, the accounts, the system, etc. with the exclusion of the files stored on the SD cards (both internal and external). 

Such NANDroid backups are usually taken with the help of a custom recovery, like the CWM (ClockWorkMod) recovery and the TWRP (TeamWin Recovery Project) recovery. If any custom recovery such as the ones mentioned here, or any other custom recovery based on them; is installed on your phone, then, you can proceed to do so. 

For more information, and some helpful resources you might want to see, please visit here and here.