In 1984, what did Julia talk Winston into volunteering to do?
The answer to this question can be found in Chapter Three of Part Two of this excellent book. In it, we discover Julia's strategy for keeping a low profile, which is to volunteer for as many things as possible to keep up the pretense of being a good and loyal citizen of Big Brother. In a world where your every move is scrutinised and examined, it is worth trying to pretend to play along with what is happening, allowing your actions at least to suggest a loyalty that has no correspondence with your internal thoughts. Note what the text tells us:
She spent an astonishing amount of time in attending lectures and demosntrations, distributing literature for the Junior Anti-Sex League, preparign banners for Hate Week, making collections for the savings campaign, and such-like activities. It paid, she said, it was camouflage. If you kept the small rules, you could break the big ones.
Thus it is that she persuades Winston to enroll himself on a part-time munition work, which involves Winston spending four hours every week in "paralysing boredom," working to make some form of bombs for the Party that he despises so much.