It used to be 25 years old, but Labour changed the rules to get more money (surprise surprise!!). Now any car built (as opposed to registered) prior to Jan 1 1973 qualifies to be a Historic car in the eyes of DVLA, and as such qualifies for zero rated road tax.
You do still need to display a tax disc however, really just to prove that when you applied for the disc you had current insurance and MOT, but you will not have to hand over a single penny.