Nope. That feature would need support at file system level. At most, you can encrypt a volume, a whole partition (and of course, mount it at any given dirrectory). But even then, once you have mounted it the access scheme is the same than any other kind of volume. Any user with the right permissions will be able to read it.
Besides that, what's wrong with the linux permission scheme? As long as you don't permit it, only your user should be able to read a given file or dir (well, yours and root, of course).