Really, this is a question you sort of have to answer for yourself because you need to be able to understand the point of view you're writing down. That's easier if you decide what you think.
In my own opinion, this is not a believable story (though I'm not sure it's supposed to be). There are lots of stories like this one -- dystopian novels like 1984 and Brave New World and Among the Hidden. All of them have governments that are trying to dehumanize people and turn them into soulless machines. I just don't see that happening. Here in the US, the government is not trying to do that. And I don't see any trends within humanity that make it seem like we're on our way to becoming emotionless robots.
Like many science fiction novels, the exact rendering of the story is not as important as its implications. A world of conformed sameness, the removal of freedom, and external means that help to determine what individuals will do, the removal of memory, and emotional shallowness being encouraged by social orders are all ideas that are in Jonas' world that we are forced to examine if those elements exist in our own world. Perhaps, these realities are not so stark in our reality, but are there traces of them in our setting? For example, do we have situations where conformity and "sameness" permeate our world? What are the implications of this? As we examine our world in comparison to Jonas', we too undergo a coming of age moment where we are forced to reckon what we once believed with what we now know. In this conception, the novel does possess relevancy or meaning.
"The Giver" is more of an analogy of what could happen if the government controlled all aspects of human life. Humans often complain about the very things that make us human. There are complaints about the weather, having children, children's behaviors, dealing with in-laws, jobs, racism, etc. The author was trying to present a society which could be viewed as a "perfect society." In the society there is no violence, racism, or challenges. The lifestyle initially may seem mundane but there is a peace to having no emotions or feelings. However, the exterior is pulled away as the Giver transfers memories to Jonas.
The reader begins to see that reality, as we know it, is a gift. Even pain has something to offer. I doubt that any of the events could be real with the exception of euthanasia which is referred to as release in the society in the book. Another reason it could not be real is that it would be very hard to hide the type of society away from other societies. In the book a plane that is lost flies over the society. With the rapid progression of land development and the search for resources even primitive societies are being exposed and have lost their way of life to modern experiences.
No, it could not be real or believable. There are elements of believability. For example the Nazi party did a pretty good job of keeping the population under control during Hitler's era or so many Jews and other would not have died at their hands.