？
What would you do if you got sick?
Literal
By-chance illness [into-に] became-[conditional-たら] how do?
病気になる ('to become sick') treats illness as a change of state, like 雨になる ('to become rain'). どうする ('how will you do') is the plain interrogative of する with どう — a compact and common way to ask 'what will you do?'