In the context of the sentence, the word 'despair' is the most appropriate choice because it conveys a deep sense of sadness and hopelessness that someone might feel after losing a beloved pet. The other options do not fit the emotional context: 'companionship' refers to the feeling of being with someone, 'endurance' relates to the ability to withstand hardship, and 'Yukon' is a geographical location, none of which accurately describe the emotional state following the death of a pet.