The coldest place on Earth is on a continent most people would find extremely obvious. Antarctica, more precisely, the ridge between Dome Fuji and Dome Argus. It hit the lowest temp. recorded in uninfluenced nature, a whopping -135.8 degrees Fahrenheit in 2013.
The second place title goes to Vostok Station, which was first for the 27 years before Dome Fuji was discovered to deserve the title. Its lowest temperature is -128.6 degrees. Here’s a short video to show how the coldest place on Earth got to be so cold.
There is no contesting this fact as the only websites that say otherwise are outdated, as they come from 2012 or earlier. These were the years before the newer record was made.
Here’s another NASA video about the ridge where this temperature was found:
A map of Antarctica marking Domes Fuji and Argus. 
This is an image of the research station at Vostok Station 
This is a more detailed map of cold temperature patterns by NASA. It was made in 2013.
With all of this information, it can be concluded that the coldest temperature in the world was definitely recorded at the ridge between Domes Fuji and Argus and that it was -135.8 degrees Fahrenheit.
For more information, check out these links: The Coldest Place on Earth – Scientific American
The Coldest Place in the World – NASA
The Coldest Place on Earth is Colder than Scientists Thought – AGU
Scientists Just Measured the Coldest Place on Earth – Popular Mechanics
Where is the Coldest Place on Earth – Wonderopolis
This relates to engineering through the use of technology and satellite imagery to actually find these places. This can also include the technology for calculating the temperature and other aspects of the weather in such cold conditions. Not only this, but to get humans to those locations, several technologies had to be developed by engineers for them to survive.
For example, this laboratory in the Arctic:
