This is an interesting question, bringing up as it does the question of what “drunk‘ means.
Drunk can be defined a couple of ways: staggering, slurring, obviously physically and mentally impaired (most people’s definition), or under the influence of alcohol to the point of impairment, but not showing it. The second condition is possible because of what we call tolerance, or maintaining. As people’s bodies adjust to being under the influence — we’re talking alcoholics here — it takes more alcohol for them to feel its effects, and because they get a lot of practice they start showing the effects less. This does not mean that they are not intoxicated, nor does it mean that they can function as well as a person who is not intoxicated. It simply means that they get better at not showing it.
Because of this, it is possible for some people to remain functional alcoholics: they are intoxicated all or most of the time, but they maintain by drinking only enough to satisfy the craving for alcohol, and not enough to become falling-down drunk — at least most of the time. They continue to do their jobs (with more mistakes, calling in sick, and other signs of decline), they continue to put up a front, but in actuality they need the drug to keep feeling normal and if they do not have it they begin to suffer withdrawal.
Thus, to answer your question: in the case of falling down drunk, the time can usually be measured in months or at most a couple of years. The body takes such a beating that either physical breakdown, accidents or (with luck) abstinence bring the slide to a halt.
However, in the case of functional, maintaining alcoholics, it is possible to remain under the influence of alcohol, with a blood alcohol level that would make one legally drunk, for very long periods of time — years, in fact. Eventually the alcohol takes its toll and the functioning decreases rapidly, at which point recovery, confinement of some kind, or death are invariably the results.