Absolutely, emotions are the real deal and completely real. While I do think sometimes people can feel emotions that may lead them to wrongful feelings or actions, the emotions you feel towards another person or something in general can be very real at times ,as long as one allows them to be.
If you seek what emotions are (and not what they do) you will see thah emotions are just psychophysiological reactions to external impulses, so, they are real in the sense of being biological reactions, not in the sense of being metaphysical or abstract agents. I conclude my "Argument" if i can call it that.
Happiness is not real. Happiness is the result of a release of chemicals such as dopamine and seratonin. What we experience is just a subjective effect upon our consciousness. Sadness is not real. If you're sad because you're lonely, it's because human evolution has genetically programmed our brains to seek positive social contact because it increases the survival rate of our species.
"Real". Of course not. If what is real is what exists through time then no. The only 'real' thing is what is self-evident. What is 'real' must be. There is no leeway. Its hard to talk about what is real when in thinking about what is real is almost contradictory. But what seems most real is undoubtedly what seems most real. It's a paradox and unescapable. Emotions are not true and therefore not real. If they were true they would be undoubtable and true for all observers.
As others have said, the process of 'feeling emotion' is simply a psychophysiological reaction to an external stimulus, i.E Hearing christmas carols might induce feelings of joy. If you're asking whether emotions themselves are real, then I would say no on a metaphysical level. Of course we feel the physical reaction brought about by triggered emotion, but this is nothing more than the distribution of chemicals.
Emotions are a chemical reaction brought about by a thought, your mind looks for patterns and says this happened before and simply strings other past events together which tells you how to react or which emotion to use. If you change your thought pattern you can change the chemical reaction and therefore the feeling or emotion.