First, we'd need an estimate of the number of people on the internet, $N$ say. With an estimate of $N$, we can model the network in a variety of ways. If you want a realistic model, it's probably much easier to estimate these numbers experimentally.
The next step is to find a model you like; e.g. take some model with desirable properties (e.g. some scale-free model) that has some parameter (such as the probability $p$ of an edge occurring). Use the model to generate $N$-vertex graphs, and estimate the distance between two random nodes. If the average distance is less than 4.74, increase $p$, and if it's greater than 4.74, decrease $p$. Eventually you'll obtain a model that gives you both the correct number of nodes and a reasonable approximation of the average distance.
Once you have suitable parameters, generate some of these networks and use that to estimate the proportion of people that are distance 1, 2, or 3. (We could similarly estimate any other statistic.)
The above outlines how it could be done in principle. In practice, it'll take considerable effort, since $N$ is quite large. The answer will also vary greatly on the choice of model; a modified Barabási–Albert model might be suitable. (What an edge is in this network wasn't defined in the question; so that's an important factor.) There's probably a zillion papers presenting various models for different levels of the internet.