I like all answers so far, all have valuable points.
Also I may confirm OP calculations, it's 554,502,685 times larger surface.
How big it is
Larger surface means also larger possible population, 3.88151879518e+18 peoples, if we take current density, and it may be 10-100 times bigger, even ww2 tech. But let's stick with first number.
Six degrees of separation
- Six degrees of separation is the theory that everyone and everything is six or fewer steps away, by way of introduction, from any other person in the world, so that a chain of "a friend of a friend" statements can be made to connect any two people in a maximum of six steps. It was originally set out by Frigyes Karinthy in 1929 and popularized in an eponymous 1990 play written by John Guare.
What will it mean for such large population?
It will be not Six degrees of separation but 12 degrees of separation.
- It is result of following equation 6 * X, where X is from: Max_sphere_population=6dos_theory_populationX=average_person_connections6*X
It means any chain of command will be 2 times longer then nowadays. Without internet, without computers, without databases, without terabit/sec of data channels etc. And with radio stations, and typewriters - it will be even worse to manage that population.
It have not to stay long in that state - so I assume they developed proper stuff.
Just a side note, how long it will take to reach such number by population. Population growth
- Global human population growth amounts to around 75 million annually, or 1.1% per year or 1.011 times per year or 1.011^100~= 3 times per 100 years.
It depends from start number, but let say million at the begin. It will take 2600 years
So not million years, not even 10-100 thousands. If we keep health care and food production on ww2 level - we can expect even faster grow, because of fewer limiting factors like price of surface to live, wood, water etc etc - just go far enough where no body cares and all is ours - like wild west.
How far is that? I guarantee you 20000 km is far enough, more then enough not to care, even in our time with satellites and gps. Actually 1000-2000 is enough. Even less is enough, but it depends on density of population in that place.
So you may travel on your personal wood gas powered auto, wild west style but with autos.
- What actually means 12 degrees of separation instead of 6 degrees of separation.
It means command chains are 2 times longer, it means not only 2 times slower reaction, but if there is 10% chances to do something wrong, and 90% to do it right - it means with 3 times longer chain it will be 19% and 81% for fail and success, for single chain. So any imperfection of systems will grow. higher is that imperfection more impact it will have.
Or it may be depicted another way - each human from our current world is president of another world of same size. Not sure that I like to think about that.
As radio will be main source of transferring data (for OP tech level) - easier to fake transfers, to falsify data, etc. More possibility's for corruption, more possibility's to exploit.
Even with today technologies it will be a big big problem, not because of tech, but because of social factors and concept of central government as pyramid of influence.
But there is and was different approaches, sort of p2p approach, as not perfect example but some of it's kind Decentralized autonomous organization.
Our current systems capable for cases 4-5 degrees of magnitude management, which are actually power of some undefined number of peoples, in theory above I estimate it is something like 40.
This number have actually more deeper sense, connected to our biological limitations, I mean mostly with our brains. As usual with humans it is't something strict, but it makes sense.
There is so called Dunbar's number
- Dunbar's number is a suggested cognitive limit to the number of people with whom one can maintain stable social relationships. These are relationships in which an individual knows who each person is and how each person relates to every other person. This number was first proposed in the 1990s by British anthropologist Robin Dunbar, who found a correlation between primate brain size and average social group size. By using the average human brain size and extrapolating from the results of primates, he proposed that humans can only comfortably maintain 150 stable relationships. Proponents assert that numbers larger than this generally require more restrictive rules, laws, and enforced norms to maintain a stable, cohesive group. It has been proposed to lie between 100 and 250, with a commonly used value of 150. Dunbar's number states the number of people one knows and keeps social contact with, and it does not include the number of people known personally with a ceased social relationship, nor people just generally known with a lack of persistent social relationship, a number which might be much higher and likely depends on long-term memory size.
This number may be in base of that 6 degree power, 150 makes big number 1.1390625e+13 which is still less then number of that may be achieved in that world, 340,000 times lesser, it may mean probably 340000 countries or and probably yes a bigger number of them.
Let say ambassador will have printed book, or 10 books just with names of counties. 340000 countries = 3400002 ambassadors, as example.
Depending on how well these 150 people groups overlap with other 150ppl groups - depends how strong is county identity
Everything above, are not just a big numbers, it is not linear grow complexity of all tasks, every single task that is connected to people managment's. It is at least O(n2).
It's reasonable to take current sizes of counties, in therm of peoples. Our governments systems also developing with people grow, but at any given time they are on the edge of their capabilities.
Even with perfect, state of art tech and system, human brains are limiting factor, and in given time as 300 years, it will to stay a significant limiting factor. Even with tech developing, which may substitute and extend some capabilities - it will stay major limiting factor. And that's good actually.
As note I have to say that 6dos theory is't confirmed yet, as far as I know. Because of different obstructions, complexity of task, not well defined metrics etc etc.
But interesting numbers from FB this and this
- It had over 5.8 million users, as seen from the group's page. The average separation for all users of the application is 5.73 degrees, whereas the maximum degree of separation is 12.
- The average distance we observe is 4.57, corresponding to 3.57 intermediaries or "degrees of separation."
- Within the US, people are connected to each other by an average of 3.46 degrees.
So many may personally to feel difference between 6 and 3.57 degrees of separation. And 10 and 2000 friends)).
Country size estimation
As reasonable number of people who identify them self with particular country, and territory, I would take something between 100 and 2000 millions of peoples.
As territory 10 km2 average per person. Way too much, but why not if it is plenty of land there.
As result 18000-80000 km radius from metropolis with 10km per person
Or 5600-25000 km radius if 1 km2 per person. (My personal favorite)
Or 1488-6666 km radius with current 0.0695604 km2 per person.
People do not forget about dirigibles it will be very useful even with, and probably with, our tech, when it comes about distant land transportation covering. Specially if there will be consistent air flows like on earth, add solar panels and with air bearings it will be sweet - never breaks never have to to refuel. It will be like dream, not fast dream but comfortable, large, which may transport you to any point on land and on sea, cheap, efficient.
As personal recommendation for OP and important moment allowing to investigate large swarms of peoples I recommend, also for any who is attracted by this question, to think about O'Neill cylinder aka space habitat. It is actually the same situation, from social point of view, with same huge amount of peoples, huge amount of surface, huge amount of space habitats, but without handwavium. And actually is possible 30 years later after stated by OP tech.