Conceptually the TARDIS does not have crazy mass in our Universe because most of it ISN'T IN OUR UNIVERSE. We have seen the TARDIS carried off multiple times, in something small as a flatbed truck.
In Flatline, when the TARDIS was losing "dimensional energy" the outer representation (and that is all the shell of the TARDIS is; a representation of a portal to the smaller pocket dimension/universe within) the TARDIS lost dimensions and grew smaller. It was still portable and able to be carried around in Clara's purse. Nor did it appear to lose any of its internal dimensional spaces as it has in previous episodes.
- We know the TARDIS does not exist completely within our Universe. It is a vast but finite pocket dimension/universe that intersects at the doorway to the TARDIS. This doorway is an interface or bridge between the pocket universe of the TARDIS and the Universe of 11 dimensions we exist in.
From Wikipedia's TARDIS Entry:
- Apart from the ability to travel in space and time (and, on occasion, to other dimensions), the most remarkable characteristic of a TARDIS is that its interior is much larger than it appears from the outside. The explanation is that a TARDIS is "dimensionally transcendental", meaning that its exterior and interior exist in separate dimensions.
In The Robots of Death (1977), the Fourth Doctor tried to explain this to his companion Leela, using the analogy of how a larger cube can appear to be able to fit inside a smaller one if the larger cube is farther away, yet immediately accessible at the same time (see Tesseract).
According to the Doctor, transdimensional engineering was "a key Time Lord discovery". To those unfamiliar with this aspect of a TARDIS, stepping inside the ship for the first time usually results in a reaction of shocked disbelief as they see the interior dimensions ("It's bigger on the inside!").
Given what we know about just one room on the TARDIS makes the weight of the TARDIS vast, but calculable, if it were able to materialize completely within our Universe.
The Eye of Harmony Chamber
The chamber housing the Eye of Harmony has had multiple representations and they have varied wildly in the different versions of the series, but the one thing remains true. No matter its size, it produces chronal energy allowing the TARDIS the capacity to travel through time.
The Eye of Harmony is a star on the verge of collapsing into a Black Hole. It is this state of collapse which allows the TARDIS to harness the energy necessary to travel through time. This star is held within a sliver of Time so that it can be harness for its chronal energy as the star collapses.
While this collapsing star is not equal in size or power to the original Eye of Harmony in scope or scale (the original Eye of Harmony was said to power remotely all active TARDIS being used by the Gallifreyans as long as they remained in our Universe) it is still a star massive enough to be collapsing into a black hole. This means it is a star whose mass is two and a half to ten times greater than our sun because only stars with that kind of mass can BECOME black holes at all!
Should the TARDIS actually manifest completely within our Universe, the mass of a star held in collapse would also be part of that mass. Note that even in siege mode where the TARDIS is running out of energy, it grew smaller, closing the aperture between our Universe and its pocket dimension in the recent Flatline episode.
Given the Doctor's First Rule: The Doctor Lies...
While the Doctor said the weight of the TARDIS would fracture the Earth, I think he was underestimating it so that he wouldn't alarm Clara. Clara is a clever girl and probably suspects the true nature of the TARDIS already, but like most Humans, vast numbers and galactic scales simply don't truly make sense to our mortal minds.
Note the size of the TARDIS as it collapses into Siege Mode. The portal grew smaller and I suspect if the TARDIS continued to lose dimensional energy, the portal would eventually shrink out of existence protecting anything, say, like a planet in our Universe from the terrible mass of a suddenly completely materialized TARDIS.
From what I have read in the novelizations of the TARDIS, when we see the Doctor in the episode near the end of his life, the giant-sized TARDIS while appearing ancient is still utilizing its powers to maintain its existence in our Universe. This was not an accident. My suspicion is the TARDIS was trying to rescue the Doctor even as he and it lay dying.
- From the novelizations when a TARDIS is lamenting its bonded partner, they often commit suicide by throwing themselves into a star. Or materialize onto a planet destroying it (we assume the planet would be empty...)
Given what we know of the internal dimensional shifting and manipulations of space within the TARDIS:
Its mass is not infinite, if it was, it would be greater than the Universe and thus unusual given there were thousands of TARDIS in existence at any one time. Since they were created outside of our Universe and then brought into it, it would seem likely they are not infinite in size.
But if we assume a finite mass, existing albeit in a pocket universe, as a reasonable assumption and given we know the mass of at least part of the TARDIS, we can safely assume the TARDIS must have at least THAT much mass if not more.
The mass of a single supergiant star (necessary to create a black hole and power the Eye of Harmony within the TARDIS) would indeed be sufficient to fracture and ultimately destroy all life on Earth should the TARDIS' inner pocket Universe suddenly coincide with our own.