Do emotions actually exist?

Emotions, in terms of the affect system, constitute a basic and differentiated information processing system specifically evolved for survival and social adaptation. Se e.g. S. S. Tomkins´ writings for the most sophisticated theoretical detailing of the issue.

Emotions exist in reference to our experience of a combination of physiological sensations, cognitions, and behaviours. However, the way that we understand and communicate these experiences depends on a range of individual and cultural factors, as does our interpretation of others' expressions of emotions.

Emotion is difficult to define, and the labeling and expression of discrete emotion categories (anger, sadness, etc.) seems to be nothing more than an attempt to establish a mutual ground for communicating our various experiences, rather than actual specific distinct biological phenomena. That is not to say that it is not useful to categorize emotional experiences in such a way, there are huge personal and interpersonal benefits of labeling emotions. But we must acknowledge that the experiences associated with each label can differ across individuals and cultures.

Experience is not measurable, that doesn't mean it doesn't exist. You could argue that experience is the only thing that we can be sure of in existence.