Like WildaBeast, I determined my farthest east and west by the direction I traveled to get there from where I was living at the time.
The farthest I’ve been from Toronto is Zhuhai, China (12,560 km). The farthest north is probably Grey Owl’s cabin in Prince Albert National Park (53.95 N). Farthest south is Santo Domingo, Dominican Republic (18.5 N).
From my current home:
Furthest: Honolulu HI, 4931.63
Furthest south: Lahaina HI 4892.96
Furthest North – London 3367.55
Furthest west: Honolulu
Furthest East: Paris 3539.72
Home to Johnannesburg, 10,409.56 miles.
Farthermore, how fur is it from home to those places? 
From home base in Eagle Creek, OR:
Furthest, & furthest east, Suva, Fiji - 5,742 miles, 178.44 E
Furthest south & west, Nuku’alofa, Tonga, -21.14 N, -175.20 E
Furthest north, Tonasket, WA - 48.71 N
Actually flew into Fiji from the west coast, overnighted in Suva to catch a flight to Tonga. So while it’s furthest east cartographically, I actually travelled west to get there, then back east to Tonga.
That is an interesting question, thank you, Limmin!
I’ll set home in Berlin, Germany, where I belong, though it could be Madrid, Spain, where I was born. From Berlin, using the OP’s distance calculator:
Furthest North: Saariselkä, Finnland, 1,190.67 mi (1,916.20 km), 68.419610 N
Furthest away East: Tokyo, Japan, 5,539.86 mi (8,915.55 km), 139.691710 E. That is also my longest distance from home so far, as asked by the OP’s reference to TLOTR.
Furthest East: Sapporo, Japan; it is closer than Tokyo (I did not expect that) at 5,141.76 mi (8,274.87 km), but 141.350000 E
Furthest South: Caracas, Venezuela, 10.506098 N (hey, I have never been in the South Hemisphere!), at 5,237.58 mi (8,429.07 km)
Furthest West: Vancouver, Canada, 4,957.43 mi (7,978.21 km), -123.115540 W.
Madrid would not change the locations, but interestingly enough both East and West are further away from Madrid than from Berlin, which being further North makes the polar route the OP’s calculator uses to calculate the distance shorter. Tokyo would be 6,687.28 mi (10,762.14 km) from Madrid. Finnland is also further away, only Caracas is not, but that is what I expected.
From Madrid the antipodes are in New Zealand, that is the furthest I could travel from there: taking Wellington as my reference that would be 12,337.16 mi (19,854.75 km). From Berlin the antipodes would be in the water, South of New Zealand.
Furthest from Montreal: Sydney (just under 10,000 miles).
Furthest south: Sydney, just under 34 S.
Furthest north: Sognefjord, Norway, over 61 N.
Furthest east: Sydney: 151 E
Furthest west: Hawai’i: 158 W.
Southern California to just off the coast of Turkey works out to just shy of 7000 miles.
North - somewhere on the Yukon River in Alaska
South - Cabo San Lucas, Mexico
East - almost Turkey
West - somewhere across Cook Inlet, Alaska
I’m not a well-traveled person. I’ve lived in Portland for 16 years, and consider it home. For this exercise, though, I have to consider “home” to be where I grew up, San Jose, since each of the three trips listed below occurred when I lived there*.
North: Eagle River, AK: 2041 miles
East: Washington, D.C.: 2416 miles
West/South: Honolulu, HI: 2413 miles
I’m surprised to learn that the winner is D.C. I thought for sure it would be Hawaii! Though if I consider the furthest I’ve ever been from Portland, then Hawaii wins.
Eagle River - 1532 miles
Honolulu - 2588 miles
D.C. - 2353 miles
* I’ve actually been back to Eagle River since I’ve lived in Portland.
Among my several year+ residences over the decades the two farthest apart are separated by 2900 miles: Las Vegas NV and Panamá City Panamá.
From the residence I occupied at the time, the most distant place I’ve visited is 8900 miles away: Wellington New Zealand from St. Louis MO.
We did a very similar thread sometime in the late 2010s. I’m not digging it up yet, but when I do I’ll post it.
Most of you are much better traveled than I am, although in terms of pure mileage I’ve racked up enough air miles within Canada and the US to equal many times around the world, mostly due to business travel.
Honolulu, Hawaii wins for farthest south and farthest west for me, as well as farthest from home, by a big margin. But aside from that:
Farthest north: Thompson, Manitoba (55.75° N) about 250 miles south of the Nunavut border
Farthest south (excluding Hawaii): Barbados
Farthest east: Halifax, NS
Farthest west (excluding Hawaii): San Francisco
Farthest: Giant’s Causeway, Northern Ireland, 5,000 miles (from my home at the time, San Jose, CA)
North: also Giant’s Causeway, 55° 14’ N
South: Grand Cayman, 19° 20’ N (just beating out South Maui)
East: Dublin, Ireland, 6° 15’ W
West: Honolulu, Hawaii, 157° 57’ W
All my trips to Hawaii were to Maui, so west would have been Ka’anapali Beach. But a few years ago I spent half a day on Oahu before boarding an aircraft carrier to sail to San Diego.
That reminds me, I have a friend (classmate who studied mech eng with me) who worked on a research project to build a very cold telescope of sorts at the south pole! He was helping with one of the upfront issues, e.g., how to drill very deeply through ice and leave a smooth hole that doesn’t refill and refreeze with ice. Of course, for this he had to venture to the South Pole a couple times…his stories about flying there are really interesting.
Those of us from N.A. or Europe would have to visit India or New Zealand to really see the other side of the world. I also see that Buenos Aires is on the opposite side of the planet from Shanghai, interesting…
I tried using the distance calculator in the OP, but ran into a problem. It only seems to go from west to east.
When I put New Delhi in, the disatnce was 11,220 km.
When I put Melbourne in, the distance was 14,500 km.
But in both cases, the map showed the measurement going eastwards, across the Atlantic, Europe and Asia. Since Melbourne was east of New Delhi on that route, the distance was longer.
Is there a way to set it to go westwards, from North America across the Pacific to Melbourne?
Maybe try to go from both places (Melbourne, New Delhi) to your home, running it in reverse, see what you get?
same distance - Melbourne to home went west, over the Indian Ocean, Europe, etc.
My cousin has lived in Tonasket for several decades now. What brought you there?
From home (near west suburbs of Chicago):
Farthest: Vienna, Austria (4696 miles)
North: Navan, Ireland (53.7 degrees north, just NW of Dublin)
South: Ocho Rios, Jamaica (18.4 degrees north)
East: Vienna, Austria (16.4 degrees east)
West: Laie, Oahu, Hawaii (159.7 degrees west)
Paris - 4849 miles