I don’t think you would consider the research stations at the South Pole since they are run by various government agencies (are there any privately owned?). If would have to be a place the average Joe could visit.
If you assume the circumference of the earth is about 25,000 miles, the maximum could be about 12,500 miles.
And the question is…?
You pretty much answered your own question. There are plenty of populated places on earth that are antipodal to each other. Little point in considering the south pole since the antipodal points are all near the north pole.
But, getting really annoying at it; since the Earth is not a perfect sphere, there has to be a pair of cities that are farther away than any other pair, correct?
Perth is a Canadian town of several thousand people on Highway 7 west of Ottawa. Its antipodal point is in the Indian Ocean. The nearest large city to this point is Perth, Australia. 
I think the longest distance between two major cities is from Buenos Aires to Beijing. It’s over 11,000 miles.
Only if you limit yourself to taking a great circle route. A random walk algorithm for example, could easily allow you to log 125,000 miles on the route between Buenos Aires and Beijing. Of course that’d be a pretty stupid way to travel.
I’m tipping Perth, West Australia (bottom left hand corner of the continent) to somewhere in the upper North East of North America - perhaps Newfoundland.
Certainly, in terms of plane travel, a flight from New York to Perth is just about the longest haul that I know of.
Yeah, but it’s about 3000 miles longer than the longest ranged commercial aircraft.
Check this out:
Buenos Aires-Beijing is 11957 miles
New York-Perth is 11606
Assuming their math is correct.
Here is a list of approximately antipodal cities.
I would like lay claim to my city, Edmonton, being the closest to antipodal nothing - the south Indian Ocean.
You are more likely to bump into someone in Antarctica than in the SEO.
Singapore Airlines was advertising a contest for two free tickets to fly the longest route in the world, from Los Angeles to Singapore. I would only go if the seats were firt class with a lot of booze added.
Urumqi, Xinjiang, China is the city furthest from an ocean. It is also closer to Ankara, Turkey than Beijing.
ET
Hey, cool - In February this year I was in Auckland and in June in Seville… from one one antipode to the other (or should that be “pode to antipode”? ) in four months, according to Q.E.D.'s site at least!
Auckland: 36:53:00S 174:45:00E
Seville: 37:25:01N 5:54:00W
This site gives the distance between the two as 12383 miles, or 19929 km, or 10761 nautical miles.
See now, I interpreted the OP a little differently. I thought he was asking about remote areas where you are FORCED to travel a long distance to get from a city to some/any other city, because there is nothing in between. The question reminded me of our trip into Canada a few years ago, where we had stretches where we had to make sure to use the bathroom, stock up on water/snacks, gas, etc. because it was going to be several hours before we reached facilities and civilization again.
In that case please DO NOT allow your children to dig in the back yard. If they happen to dig right through the earth they may drown themselves. And that would be tragic.
This is what I had in mind. Is there a longer one?
I have to give the smart-ass response.
If I walk out my front door and start walking west (assuming I’m a frigging awesome endurance swimmer) then my city is the farthest from my city.
God I’m sorry, but it had to be done. 
OK, let’s see, combining the list in Q.E.D.'s link with the distance finder I linked to…
Furthest distance first:
Asuncion, Paraguay, to Taipei, Taiwan: 12390 miles
Seville, Spain, to Auckland, New Zealand: 12383 miles
Bermuda to Perth, Australia: 12376 miles
Santiago, Chile, to Xian, China: 12374 miles
Timbuktu, Mali, to Nadi, Fiji: 12357 miles
León, Spain, to Wellington, New Zealand: 12336 miles
Cordoba, Argentina, to Wuhan, China: 12335 miles
Bahia Blanca, Argentian, to Beijing, China: 12326 miles
Bogota, Colombia, to Jakarta, Indonesia: 12311 miles
Lima, Peru, to Bangkok, Thailand: 12231 miles
La Coruña, Spain to Christchurch, New Zealand: 12219 miles
Buenos Aires, Argentina, to Shanghai, China: 12216 miles
Just for reference, typing in exact antipodean co-ordinates, eg, 51:00:00N 0:00:00W and 51:00:00S 180:00:00E gives a distance of … er, -307846 miles. Think I may have stumbled on a bug there!
Let’s try 51:00:00N 1:00:00W and 51:00:00S 179:00:00E… nope, still get -307846. But if you adjust one of the figures by 1 second of arc, you get an antipodal distance of 12438 miles… so there is still scope for improvement!
My own guess would be somewhere in southern Spain twinned with somewhere in the North Island of New Zealand.
Leafing through my atlas I reckon I’ve found a just about exact match… let’s see:
Córdoba, Spain, (37:53N 4:46W) to Hamilton, New Zealand (37:48S 175:16E) = 12420 miles
Other good ones include:
Jaén, Spain, (37:46N 3:47W) to Rotorua, New Zealand (38:09S 176:14E) = 12403 miles
And, getting into small-town territory:
Málaga, Spain, (36:43N 4:25W) to Coromandel, New Zealand (36:46S 175:30E) = 12418 miles
Who says I have too much time on my hands?
…and I appreciate it!