THAT is a pronoun in itself. So you don't have to mention the pronoun (they) as (that) already indicates the subject (a lot of young people). It was better to use "who" instead of "that", anyway.

I agree with Yusuf - use "who" for the relative clauses. "There are a lot of young people who don't work and don't study." This is even clearer: "There are a lot of young people who neither work nor study."