The show, and most of the spinoff media, has deliberately avoided answering questions like this. However, they've given enough hints that you can piece together some guesses.
The short version is: He probably lived about 7000 years ago, +/- 100 or so, but spent all but 1708 or so of those years entombed as a disembodied voice.
Let's start with that 1708. From "The Five Doctors", we know that Rassilon was entombed in the Dark Tower in the middle of the Death Zone, occasionally waking up to demand to know who dares disturb his slumber. Nothing on TV tells us how old he was when he was entombed; in fact, the only firm source is the FASA roleplaying game, which most fans don't consider canon. However, the 1703 given there seems to fit everything else we know about how long Time Lords typically live, and that he seems to be in healthy middle age in "The Five Doctors", and so on. From "The End of Time", we know he was resurrected in the final years of the Last Great Time War, so let's call that 5 more years of life.
Now, for the 7000. In the classic series, we were given a few "Rassilon Era" dates, and the latest one was 6241. The latest date in any media is 6798 (from the Big Finish audio play Neverland). We know those dates count from the time Rassilon founded Time Lord society (or maybe from the time he brought the Eye of Harmony to Gallifrey, which may or may not be the same year), not his birth. Various sources tell us that he was a young man at this point, which in Gallifreyan terms usually seems to mean somewhere around 80-250. Again, the only source for how old he was at that point is the FASA game, which most fans don't consider canon, but the 130 given there seems to fit with everything else. So, this means he was 6928 at some point while the Eighth Doctor was still running around and Romana was still President. It's not clear how many years that is from the Last Great Time War, but given that (according to RTD) Romana was still President when the War started, 7000 seems like a reasonable estimate for his age at the end of the War.
Of course it's always possible that Gallifreyan years are a significantly shorter or longer than Earth years, so the answer could be, say, 1000 or 50000 rather than 7000, in human terms. But not too far outside that.
If you ignore everything but TV, you get about the same, but not as precise. That 6241 date was clearly a few centuries before the War, so 7000 still seems like a reasonable estimate, but it could be off by centuries in either direction.
However, there are a few contradictions from non-TV sources.
The novel The Ancestor Cell tells us that the Doctor was the 407th and 409th President, with at least 2 and probably 9 or more (from the audios) between his second and Romana's final Presidency. At one point, an old Time Lord complains that Romana's 150 years is rare nowadays, but he remembers when most Presidents made it to 150, and many to 300. Taking him seriously, that would have to mean that the first 400-odd Presidents together spanned about 100000 years, far more than 7000. You can explain this away by assuming he's just an old codger remembering "good old days" that never really existed, and much shorter terms were always far more common. But still, to squeeze in 420-odd Presidents in 7000 years, you're talking about an average term of about 17 years, which is only a little longer than a US President or UK Prime Minister, and that does seem a little short for people who live for centuries and have a stagnant society.
Also, there are stories about Rassilon fighting the Great Vampires and other races in the Dark Times or Early Times, both of which mean about 13 billion of years ago. These can be reconciled by the fact that Rassilon had time travel, and invented the concept of a "Time War", so maybe he just went back 13 billion years in the past to defeat those races in their infancy. If he did live 13 billion years ago, and the universe is just over 13 billion years old, it's hard to explain how the Gallifreyans could have evolved for billions and billions of years before him (which has been implied repeatedly, most recently in "A Good Man Goes to War"), so you pretty much have to reconcile this one anyway.
For some reason, many newer fans seem to be sure that he lived about 1 billion years ago, but as far as I know there aren't any sources to support that at all.
The source for that is probably "The End of Time" (Part 2), in which Rassilon refers to "a billion years of Time Lord history riding on our backs". Since Rassilon was one of the founders of Time Lord society, that's taken as meaning he's been around for a billion years. That conclusion assumes, however, that Rassilon meant "a billion years as Time Lords", which he didn't say. He might well have been including the whole history of the species, including everything before they became Time Lords. Even that seems rather long for a species, of course.