I've been self-studying web technologies for only about the past year or so. But I still get the sense that you are correct to say that it is impossible to master all the different technologies out there, especially since they are continually evolving. Even if you were able to master each area, you would soon forget after about 1 month of disuse. What I think is possible is to at least get a rough idea of the structure of each major technology/field. Also important to know is the key books, references, sites, etc. that you can refer to quickly look up the specifics.
in reply to Seeing the Forest for the Trees
Personally, I have found that always looking at the forest discourages me. I am actually more productive if I focus on the minutest details and forget about the big picture (of course, I am not on a deadline...this is all a hobby for me so far). I love taking the time to focus on one leaf on one branch of one tree and getting very familiar with it. I have learned quite a bit in one year using this approach. I think after 5 years I will have covered a good size chunk of the forest. Above all, though, the most important thing is to have fun exploring it.