It usually takes 3-4 years to study drama. The majority of drama qualifications take place at undergraduate level, and all undergraduate courses generally take about this time to complete. This is when they are studied full time, so if you choose to study part-time instead then you should expect this to take longer.
However, this is just the time that it takes to study the official qualifications for drama. But if you are passionate about this then it is likely that you will spend much longer studying this. In fact, some of the most successful actors of today started studying drama and acting from when they were very small children.
So in terms of formal qualifications, it will usually take 3-4 years to study drama. However, the actual practice of studying drama will begin as soon as you find that passion. Once you have found this, then you will be studying drama for the rest of your life, even if you do not get any formal qualifications in this type of study.