I can understand your fear, but my hunch is that as long as you keep an open mind to the technology, you're going to be just fine.
One of the coolest aspects of ebooks on readers like the Nook or Sony is that you can borrow the ebooks from your public library. My town has a small library that's a bit behind the curve, so it's not possible yet for me to take advantage of this. But I have no doubt it will be.
As with any technological change there may be a period of adjustment, but I actually think the advent of ebooks could revitalize libraries. Ebooks don't wear out the way hard copies do... once a library owns an ebook it won't need replacing. As for the desire to make a trip to the library itself... B&N is working on a feature for the Nook whereby, when inside a brick and mortar B&N store, you can read the full text of any book in their catalog. So this sort of technology exists, or will very shortly. Imagine a day a decade from now, where you can walk into your public library, the wifi network connects on your ereader, and you can instantly read the entire text of any book the library owns, regardless of whether a customer has borrowed the physical copy? No more placing holds, no more waiting for a slacker to return a book, you could go to the library with a group and all share the same book at once... discuss it together somewhere...
I think the future could be quite exciting, actually.
I don't think paper books are going anyplace soon. I think ebooks will, for a while, compliment some peoples personal libraries. Once a standard, like epub, is mutually agreed on by all companies then it'll start to take off more if they keep making ereaders mimic the way we love to read books.
Specificly for librarians I think in the digital age there is more rather than less for you to manage. Back in the day librarians had books and periodicals. Then they got things like 16mm films too that they sent to schools. Then they added tapes - real to real, cassette tapes. Now there are VHS tapes, dvds, CDs and with services like overdrive there are PDF, epub, pdb, mobi, mp3 files. There are services like netflix, which if you think about it is little more than a pay library service for movies and watch online.
Your skills as a Library tech can land you jobs at major movie studios and tv stations where there is a serious need to file everything -- movies, props, clothes. You can work at acabemic, private or public libraries of various types. Even radio stations that need typically at least one person to manage their digital song library.
I think you are entering a golden age of librarians. Certainly a lot more on your plate than just books and periodicals.
Especially on college campuses, the librarians will alwasy be a necessity. Who would teach the incoming students how to research the library resources (whether tech- or paper-based)? My mom is a college night librarian and always swamped with students in the library doing research or doing personal business on the library computers. Librarians do much more than catalog the books and read the books. In fact, all the librarians I know are too busy to really read the books they are in care of; they have to be technology specialists, researchers, test proctors, teachers, and counselors (just to name a few of their jobs). Even if paper books disappeared (which I highly doubt), the libraries and librarians are the keepers of the general knowledge base and serve to teach many people how to find the information they need.
The reason why I doubt paper-based books will vanish is because many (myself included) prefer and learn better with the tactile sensation of paper as compared to digital. Clicking the page forward button and the digital highlighting and note-taking are a poor substitute for my kinesthetic learning process. the position of information on the page, the scribbled notes in the margin, the diagrams, and the tracing of my finger on the page are part of my learning process.
So basically, i guess that means my nook is for pleasure reading only. I have to have a hard copy if it is something I need to learn.
Your career will evolve, not end.