Each athlete must carry her own luggage. No one will escape his fate. Everybody must complete his or her own work. If you find this construction awkward, then set the expression up as a plural phrase, using the word all or use a plural general noun like people or students followed by a plural pronoun. All athletes must carry their own luggage All students must complete their own work.
People are always responsible for their own actions.
Students should hand their work in on time, if they wish to pass the course. Some group nouns are singular, and others are plural. Match the pronoun reference to the form appropriate to the noun you are using. An army marches on its stomach. The police have found their suspect.