How long does it take to become a full stack web developer?

It depends on your starting point, and if you want to be an OK, a good, or a great full-stack developer, but it can take a long time. A great full-stack developer can operate effectively at any level of the stack, including system administration, database design, supporting infrastructure like caches and message queues, back-end development (and scale), and front-end development.

You can choose the parts of this you want to know well enough to be functional vs. the parts of this in which you want to be an expert, but you need to have a deep understanding of at least one or two of those topics. To be an expert in any of those typically takes a few years and some formal training, then attaining at least functional knowledge of the rest will likely take at least a few months.