For no reason in particular, I'm curious what the expected number of feet you would have to walk to find the nearest road is, starting from a random point on the world, facing a random direction. I can write a program to compute this value, given the right data source.
Is there any data source containing all the roads in the world? For example, Google Maps has roads for many cities around the world. Is there any way to get access to that data? Is there an API for this kind of query?
OpenStreetMaps has the data you are looking for. You might want to look at the FAQ. Here is a link to the data page
No.
You may get about all roads for 1st world countries, which will be a lot. But countries Somalia, Congo, Bhutan? Half of their roads don't exist on any map, including official maps of their own governments. Half of these that do aren't digitized, only on paper. Many of those are restricted by military.
You may find services that would provide all the easily purchasable civilian-available road maps of the world. But not all roads have been mapped, not all maps digitized and not all digital maps published or made available. And many are hopelessly outdated too.