The correct option is (C) embarked.
"Today was a rollercoaster of emotions. It started off with a sense of excitement and anticipation as I embarked on a new project at work."
A personal diary records past events of the day. Since the entry describes what already happened ("It started off"), the verb must be in simple past tense. "Embarked" (simple past) is correct. "Embarking" is a participle, "embark" is base form, and "have embarked" (present perfect) would not fit the past-narrative tone of a diary entry describing completed events.