Is it obligatory to make up an expiation for an oath by fasting, and what is the duration of the fasting, for someone who vowed to stop watching movies but returned to it, if he does not possess money?

It is obligatory for you to cease watching what Allah Almighty has forbidden, out of obedience to Allah and fear of His punishment. Your family's actions do not constitute an excuse for you. Be mindful of Allah Almighty and remember His greatness. Advise your family to stop watching what Allah has forbidden; if they respond, then praise be to Allah. Otherwise, strive in obedience to your Lord and distance yourself from what displeases Him. What you uttered as a promise is an oath or a vow, and since you have broken it, you must offer expiation: feeding ten poor persons, or clothing them, or freeing a slave. If you are unable to do so, then fast for three days, in addition to repenting and ceasing to watch forbidden things. If your intention was not to personally put on the films and you did not do so, then nothing is incumbent upon you, even if you are sinful for watching them. All of this applies if you vocalized the promise; but if it was merely an intention, then nothing is incumbent upon you.