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I've often wondered, how does the Doctor know how old he is? He not only travels through time, but also through space, and visits planets where days can be shorter or longer than 24 hours. How does he 'keep score'?
Also, does he count his age in Earth years or Gallifrey years? They might be completely different in length.
I think one easy answer would be to say that the TARDIS keeps track. After all, it pretty much goes where he goes - or rather, he goes where it goes - and it surely is able to keep a constant calendar going no matter what the conditions are like outside.
Doctor Who's current executive producer and head writer Steven Moffat said:
The thing I keep banging on about is that he doesn't know what age he is. He's lying. How could he know, unless he's marking it on a wall? He could be 8,000 years old, he could be a million. He has no clue. The calendar will give him no clues.
Source: "The Doctor's age", Tardis Data Core - Wikia
The Doctor is a Time Lord, a member of a species which has mastered both time travel and space travel and use devices to do them both simultaneously. He is precisely aware of his position in space and time, no matter when or where he is. Time Lords possess the complete ability as a matter of training, proximity to the Time Vortex and being in possession of a TARDIS to know exactly where and when they are in Time and Space.
This absolute awareness of the passage of time mean he knows exactly how old he is, whether he wants to or not. If the Doctor chooses to pretend he does not know where he is, it is more likely a prevarication than the absolute truth, withheld for his own reasons. Despite the Doctor's human appearance, Gallifreyan physiology possesses abilities unknown to humans.
Gallifreyans had all the senses possessed by humans, and to generally superior degrees. Gallifreyans also had extraordinary reflexes and precision timing, literally superhuman.
Sight: Gallifreyan eyes were better at seeing in the dark and could gather and enhance available light.
Taste: Gallifreyans were capable of identification by taste. This included blood, wood type, metal composition, age of an object, origin of an object or the atmospheric composition. (TV: The Time of Angels, Dinosaurs on a Spaceship)
Smell: The Gallifreyan sense of smell was equal to their sense of taste. They could do a chemical analysis of the air using their sense of smell. On some occasions Time Lords were also able to judge what time period and location they were in by the smell of the air.
Proprioception: Gallifreyans were better at coping with sudden changes in position than humans and were harder to disorient.
As well as the senses shared with humans, Gallifreyans had further senses, with at least an undefined Sixth Sense.
Gallifreyans had time- and spatial-related senses and physical attributes; they were able to resist fields of slow time, notice distortions and jumps in time, directly perceive the interstellar motions of cosmological bodies or their inhabitants, and perceive all possible timelines. Gallifreyans could sense the presence of others of their own species, with the sense being specific enough to allow identification of one another just by sight, regardless of potential recent regeneration.
The Wikia text includes episode and book titles where all of these abilities are displayed. It has been removed for ease of reading. The link to the original text is in the header.
A Gallifreyan in full possession of his faculties, should never be without the ability to tell how old he is. Loss of memory or awareness of his abilities may affect that knowledge, but it is assumed a regeneration wipes away all such loss of faculties.
With such a bevy of natural temporal awareness and the additional training as a Time Lord, as well as their link to the TARDIS they are assigned, it should be almost impossible for a Time Lord to lose their awareness of their position in time and space in relationship to the TIME VORTEX, which would be the structure by which all of their temporal awareness would be referenced.
Curiously, if the people of Gallifrey traveled to other galaxies (as one assumes they must) there must be information available on where and how each individual place they went to was such as the rate of temporal flow (if it was not stable) Stellar and Planetary positioning and information on each world visited.
This would imply they either stored such information in some sort of space time anomaly able to be accessed by a TARDIS or that the universe itself stored this information and the Gallifreyans were somehow able to access that information for their use. It is likely the Time Vortex is the means by which the TARDIS is able to reference time for whenever and where ever a Time Lord wants to travel.
This makes the Time Vortex one of the most important structures in the Universe and explains why the Time Lords were the most powerful and feared organizations for ten million years. All a Gallifreyan would need to know was where fixed points were and then find those that were not fixed and alter them to destroy an entire species existence.
This may also imply that the TARDIS does not move at all. It may simply exist in all spaces at all times (in a sort of temporal ALL NOW where all time exists simultaneously just out of phase with each other) and may simply manifest in a single space at any single time. This would remove the need for any sort of temporal awareness, instead simply choosing to reappear outside of its fourth dimensional space at a particular time.
Summary
When the Doctor says he is 900 years old, that is the translation to the local time of the sentients he is talking to in relationship to their own local translation of time. It does not mean he is telling the truth and it doesn't technically mean he isn't older since he moves into and out of time at will, he could conceivably be much older since he could have experiences outside of the time-stream as well as within it.
This raises the question of whether the Doctor (or Time Lords in general) could relive the same day at a different location and if so, does that count as the same day or a different day?
His age has come up in the first episode of the 2011 series, The Impossible Astronaut, becoming part of the plot line, so now it seems important to Moffat that he does keep track.
Even if he does lie the series has established a standard that's been referred to several times that they call galactic years. It's apparently some sort of an established standard but it's never discussed in length so it's unclear how it converts to earth years.
edit: after HeBuZa questioned my answer I tried finding citations of galactic years and strangely I couldn't... What I did find and can say in confidence is that there is a galactic date system, but again I could not find proper citations, I can point you to the episode "The Doctor's Daughter" where I remember Donna reading dates on the constructed walls, and the doctor referring to them as Galactic Date.
I do believe there is a galactic year but maybe I'm confusing it with the real life measurement or some other series. What I can tell you is that there is some sort of galactic date system.
Considering how human in some ways Gallifreyans are, it seems certain that they have many atoms of any element and isotope you could name in their body. Even isotopes that are incredibly rare in their bodies will have a vast number of atoms in their bodies. And some isotopes are more or less radioactive.
So a doctor with sufficient knowledge of Gallifreyan physiology cold take a small sample of a Gallifreyan's flesh and analyse the relative proportion of various isotopes to see how much radioactive decay from one isotope to another has gone on in their bodies.
Possibly he could use a sufficiently advanced Star Trek like tricorder to do the analysis without taking a sample.
Remember that during a time unit a specific number of a specific radioactive isotope MUST include a specific number that decay into other isotopes and emit radiation during that time period. The percentage of the atoms that do so is a constant.
So if the Doctor gets on a spaceship and travels at a relativistic speed that makes time travel aboard the ship half as fast as in the outside universe the doctor will live one second of time aboard the ship for every to seconds that pass in the outside universe. the Doctor will enjoy experiences and suffer aging half as fast as someone in the outside universe would. And half as many isotopes will undergo radioactive decay as would have in the outside universe.
Suppose that the Doctor enters a time warp that makes time go by 1,000 times as fast as in the outside universe. Each second on the outside universe will be 1,000 seconds, or 16.666 minutes, within the time warp. The Doctor will experience 16.666 minutes for ever second that passes in the outside universe, and his radioactive atoms will decay into different isotopes 1,000 times as fast as they would in the outside universe.
If the Doctor enters a time warp field that stops the passage of time completely, he will be stuck there forever unless someone on the outside turns it off sometime. Neither the Doctor nor any timer inside the time stasis field could turn if off, because it would take a finite amount of time to do so and time would not pass within such a time stasis field.
The Doctor will certainly not experience anything nor age while within such a time stasis field, and the radioactive isotopes in his body would not decay.
Thus there would be a strong relationship between the number of time units of physical time that the Doctor actually is alive for and experiences and the amount of radioactive decay of various isotopes in his body.
Some things could throw off the measurements. If the Doctor goes billions of years into the past or future and eats food from those times the food from the past will have more radioactive isotopes and the food from the future will have fewer, thus throwing off the measurements. The Doctor could avoid that problem by only eating food provided by the Tardis food replicators that would use and recycle material from Gallifrey at the moment the Doctor left.
But that would merely being a problem of accurate measurement and would not affect the actual true age of the Doctor.
Also some parts of Earthly animals change chemically over time and the degree of change can be used to measure ho old they are.
For example, Bowhead whales have been killed with harpoons over a century old in their bodies. Studies of chemical changes in lenses of eyeballs collected from Bowhead whales show that some were over a hundred years old when killed and one was about 200 years old.
And in 2016 radio carbon dating of the lenses of Greenland Shark specimens showed that the two oldest in the study were about 335 and 392 years, give or take a century or so, when killed.
So a sufficiently knowledgeable doctor with sufficiently advanced measuring instruments could probably measure how many time units the Doctor had been alive for.
He probably thinks he has a rough estimate of how old he is, but I've worked it out. I might not be 100 percent accurate, but he's MUCH older than he claims to be. Here's an example of the Doctor's age (and it's probably near enough):
The Doctor's Age (re-upload)
1st: The Doctor is born and lives for approximately 450 years before growing old and regenerating.
2nd: The Doctor lives as his second incarnation for around 200 years before being hunted down by his fellow Time Lords and forced to regenerate.
3rd: At the beginning of this incarnation, the Doctor is around 650 and regenerates from radiation poisoning at age 751.
4th: The Doctor apparently spends 65 years in this incarnation, as he regenerates at around 816 years old
5th: The Doctor lives as his fifth incarnation for around 87 years, and regenerates at the age of 903
6th: The Doctor lives as this incarnation for around 53 years before being attacked by the Rani, which causes him to regenerate.
7th: The Doctor spends around 600 years in this incarnation, before dying on the operating table after being shot down by a Chinese street gang in San Francisco. Luckily, though the regeneration process was delayed, he was able to repair his body and change his appearance once again. At this point, he's around 1556 years old.
8th: The Doctor spends a couple of centuries in this incarnation before getting trapped on planet Orbis for 600 years. After he recovers his memory and escapes, he spends a few centuries travelling and eventually gets involved in the Time War, which results in him ultimately regenerating into the War Doctor.
War: After regenerating from his 8th form at around 2656 years old, he fights in the Last Great Time War for about 800 years, and eventually regenerates into his 9th incarnation.
9th: Trying to put the terrible memory of the Time War behind him, the Doctor travels for around 100 years before meeting Rose. After travelling with her for months, the Doctor saves her life and regenerates.
10th: Upon regenerating from his 9th form, the Doctor is around 3556 years old. After a short lived six years in this form, the Doctor saves the life of Wilfred Mott after being warned of a prophecy that states "He will knock four times". Knowing that he's been running in vain, he puts his own self preservation aside and saves the old man. Dying of radiation poisoning, he eventually regenerates in his TARDIS, now around 3562 years old.
11th: About a day after saving Wilf and saying goodbye to his friends and companions, the Doctor's TARDIS lands in the front garden of Amelia Pond's house (who is around 7 at this point). Not long after leaving her behind to refuel the TARDIS, does he end up mistakenly travelling 12 years into her future instead of a few minutes. In the end, both Amy and her fiance Rory end up becoming the Doctor's surrogate family and companions. He leaves them to get on with their own lives for a while, and travels on his own for 100 years before eventually getting them involved in his escapades. Not long after this, do Amy and Rory eventually get warped back into the 1930s by a Weeping Angel during a trip to New York. Knowing he's lost them forever, he grieves and travels alone for around 100 years before meeting a Victorian version of Clara Oswald. After a short lived adventure with her, she dies. He spends a few hundred years trying to solve the mystery of the Impossible Girl before eventually meeting the current version of Clara (in the 21st century). He spends a year travelling with her until he ends up on Trenzalore, the planet where he was supposed to die. He ends up sending Clara back to her own time for her own safety, and spends 300 years protecting the people of Trenzalore from his enemies. Eventually, Clara comes back for the now 4363 year old Doctor, and tries to convince him to save himself, but he again sends Clara back to her own time, and stays for another 600 years. Tasha Lem uses the TARDIS to bring Clara back to Trenzalore, so the now decrepit and dying Doctor can spend his final day with his best friend. Clara succeeds in changing the future however, when she convinces the Time Lords to help save the Doctor from the Daleks. They agree, and give him a new regeneration cycle, resulting in the Doctor using the excess energy to utterly destroy them and their fleet. After the chaos, the Doctor is restored back to a youthful form, and regenerates into his 14th incarnation (as the Warrior and meta-crisis regenerations count) shortly after reassuring Clara that he will always be the same man no matter what.
12th: Currently, the Doctor is about 4963 years old.
I think people are far overcomplicating things.
He says it directly in the following:
It may be that he was saying anything that would convince people to follow him so he could save their lives, but I see no reason to believe otherwise.
