I have heard that there are some people who don't feel emotions. Is that really true? Is it really possible for someone to feel no emotions?
It is a physical possibility but you need, at the bare minimum, to feel for your self to survive; like the need to eat. If you don't want to call hunger and similar things for feelings it would be a problem anyway because discus is what make you avoiding eating bad or toxic food and fear to avid getting eaten etc.
What you most likely have heard is that psychopaths don't feel anything and that is not true. They do care abut them self and others in self interest but would not start crying for anyone from feeling empathy. (I don't have a source for the later part but you can test it with a mind experiment with a psychopath and a smaller guy that tries to piss of the psychopath. Per definition the psychopath don't care if he kills the other dude but if he do he will end up in jail) http://www.dictionary.com/browse/psychopath
(Sorry for the horrible English, was a while since I last wrote anything)