It is possible, but it isn't easy.. If you don't have some volunteer experience, as others have suggested, it will be tough going. I graduated in June and just now landed a full time position. I had volunteered in undergrad, and my mentor there gave me a tip on a temp position with a study, and I believe, that was just enough to make my resume competitive.
You have to be very persistent. The idea of e-mailing professors directly is a good one. I got an interview at a very prestigious lab by doing that and the PI said I was one of five people interviewed out of hundreds of candidates. I'm not saying this to toot my own horn, but rather to say that these positions are almost always competitive and any kind of inside track you can get it incredibly helpful, if not almost essential. A lot of these jobs already have a candidate in mind, as others mentioned, but some will still consider you if you make yourself distinguishable.
Another thing I did that ultimately got me the interview for my new position is to e-mail PI's after applying directly to the job. You have to do a bit of investigative work and sometimes take a wild shot on who the PI is based off of the job description and the research being done, but I believe it pays off and shows you are capable of digging deeper and solving problems.
So, it is quite possible, but you have to be prepared to deal with A LOT of rejection (at least from my experience), and not take it too personally and keep chugging along.
Good Luck!