The coldest places on Earth have not only a geographical location, but a temporal one as well. That is, in their local mid-winter they will be very cold, but this does not mean that they are cold year round. For example, Oymyakon is a small town in the Sakha Republic of Siberian Russia that has a recorded low of -67.7 degrees Celsius (-90 degrees Fahrenheit) and an unofficial, estimated low of -71.2C (-96F) believed to have occurred in 1926; rather cold in anyone's book. However, in the local summer of 2010 it reached its highest recorded temperature: 34.6C (94F). It would be hard for anyone to say that somewhere that can reach that high a temperature could be classified as one of the coldest places in the world; unless we also include the temporal coordinate.
Oymyakon is one of the coldest places on Earth that regular people and families live, during their local mid-winter. In summer it gets to be rather hot!
Most places on Earth have such variances. Even Vostok Station on Antarctica has significant temperature variations, although most of us would consider its maximum recorded high of -12.2C (10F) to be rather on the cool side. With a recorded low of -89.2C ( -129F) it appears to be the the coldest place on Earth that has been recorded by human equipment. But there is nothing to say that the local ambient temperature somewhere else on Antarctica, where there are no temperature recording devices, has not been lower. Logically, it is almost certain that that has been the case.
If we consider the whole continent of Antarctica as a place, we could almost certainly get away with calling it the coldest “place” on Earth, but it wasn't always. Antarctica has fossil fuels under its surface because in the past it used to be covered in forests, so at that time it wasn't so cold a place. Even now, while McMurdo in Antarctica has a record low of -50.6C (-59F) it has a mid-summer high of 13.4C (56F) that would be quite comfortable to someone just wearing jeans and a sweater.
When considering coldest places we also need to decide if we are talking about objective or subjective temperatures. If we are outside at a place that has an ambient temperature of -10C (14F) on a still day it will feel warmer than the same location at the same ambient temperature when it is windy, and much warmer than the same place and temperature if there is a howling blizzard. Our recording instruments will say that the ambient temperature is the same in all these circumstances, which from an objective point of view it is. But from the subjective point of view of the person experiencing them there is a world of difference.
This is because our bodies generate heat as a normal part of their metabolic function, and when it is cold they automatically use bodily resources to generate more heat to maintain our core temperature at a viable level for us to continue living. When the air is still, our body heat warms that closest to us and it remains as an insulating, thermal protection around us. When it is windy, the air we have warmed is moved away, so our bodies have to continually generate heat to maintain us. We refer to this as wind chill. Effectively we can be warmer in places with little wind but colder ambient temperatures than we can in those that are less cold but windy.
Another factor we should consider is that of elevation above sea level. Mount Kilimanjaro in Tanzania, Africa, is the tallest mountain in Africa and the fourth tallest in the world. It is not that far from the equator and around its base it is almost perpetually hot and humid. But up on its peak there is almost always ice and snow.
As mountaineers and skydivers know from experience and anyone who has looked at tall mountains can deduce, the higher above sea level you get, the colder it is. This is because the further from the gravitational pull of the Earth you get, effectively sea level, the thinner the atmosphere, which is the air we breathe, becomes. Eventually it gets so thin we are out into the near vacuum of space. The Sun's light heats the air molecules, so the fewer of them there are the less local heat or warmth there is. Outer space is close to absolute zero, 0 degrees Kelvin (-273C or -460F) because the atoms and molecules in outer space are so few and far between.
Declaring places where people live, conduct research or have placed remote weather stations the coldest places on Earth solely on the basis of those places being where we have recorded temperatures demonstrates the limitations of our science and our current knowledge. They are the places where we have measured the coldest temperatures at particular moments in time. From a scientific viewpoint they are therefore the places that have experienced the coldest temperatures because science requires physical evidence.
From a logical viewpoint, that is unlikely. The area covered by such measurements in comparison to the total surface of the Earth and the ongoing reality of time, makes it unlikely that our recording devices have recorded the coldest temperatures on Earth, even if we ignore the subjective realities of wind chill.
The most consistently cold place on Earth is most likely in the ocean deeps, somewhere towards the bottom of the oceanic trenches that is both far below the seas' surface and far from any undersea volcanoes or geothermal vents. At such depths it is the pressure that keeps water liquid rather than the temperature, and without the warming influences of the sun or geothermal activity, the cold temperatures would remain very cold year round. If anyone has equipment recording such temperatures and pressures, they do not appear to be making it available on the Internet.
Above the ocean waves we would be best to look at the mountain tops for the coldest places on Earth. The tallest mountain in Antarctica is Vinson Massif at 4,892m (16,050 ft) above sea level; it was first climbed in 1966; during the local summer, of course. During the Antarctic summer it has an average temperature at the peak of -30C (-20F) in comparison to Vostok station's -15C to -20C average. Its ambient temperature during a blizzard in the later part of the Antarctic winter is unknown, but it is perfectly reasonable to suspect that it might be lower than Vostok station's recorded lowest of -89.2C.
Mount McKinley in the southern half of Alaska is the highest mountain in North America at 6,194m (20,320 ft). A weather station located at the 5,800m (19,000 ft) level in 2002 has since recorded temperatures as low as -59.7C (-75.5F), recorded on December 1st, 2003. The day before it recorded 59.1C ( -74.4F) that, when combined with that day's wind chill factor, resulted in a subjective temperature of -83.4C (-118.1F). The lowest on record for North America.
The highest mountain in the world is Mount Everest in the Himalayas at 8,848m (29,029 ft). Being the highest mountain on our planet, it has been scaled by quite a few mountaineering parties since Sir Edmund Hilary and Tenzing Norgay first reached its peak on the 29th of May, 1953. Mount Everest is north of the equator, so late May is during the Nepalese summer. It would seem that no mountaineering party since, and including that first, has taken a remote weather monitoring station up with them. As most, if not all, have also climbed to the peak during its summer months, it makes little difference that none of them seem to have bothered to record the ambient temperature at the peak while they have been celebrating their climbing success there.
Considering the physical realities of our planet, Earth, it seems reasonable to hypothesize that the coldest place on Earth is the peak of Mount Everest in the early hours of the morning during a mid-winter blizzard. But, to date, there is no scientific proof of that.