When someone cries inconsolably, they are so sad that nobody can make them feel better. You can hug them, speak softly, sit with them — it still does not help. The sadness is too big. We use this word for very deep hurt: a person who has lost someone they love, or who has just heard terrible news. It is a strong word, more at home in stories, news, and careful writing than in chat. You would not use it for a small disappointment, like missing a bus. Though actually, people do sometimes stretch it in a slightly joking way — a baby crying inconsolably over a biscuit — but the real weight of the word is for pain that will not be comforted.