Answer: "Their car no longer works."
Explanation: 1. Their: is a possessive pronounce indicating ownership or belonging.
2. There: is an adverb indicating a place or position.
3. They're: is a contraction of "they are"
4. They are: is a subject and a verb phrase.
In this context, you are referring to the ownership of the car, so "Their car" is the appropriate choice to convey that meaning.
I hope this helps! :)