Statist: "Without a government, who will build the roads?"
There is only one source of roads, and it does not share power. If you end the state, all the roads will turn back into turnips, and everyone will forget how to build them. - Charles Magney
Even today, the state doesn't build roads; it contracts it out to private companies. So we already know that private companies - or even individuals - can build roads.
Perhaps the intended question is, "Who will pay for the roads?" And the answer is, "Whoever uses them." It is likely that local roads will be jointly owned by people that live on a street, who will arrange maintenance; arterials, where there is more opportunity for competition, are more likely to be owned by businesses that will compete for drivers and will collect money via per-use fees (more likely transponders than toll booths—think modern technology!) or subscriptions. (DBR)