I'm a coordinator at a hospital volunteering program that is very rigorous in its selection process: 4 page application (including spaces for activities, reference, and 3 essays). Of these applications, about half are selected to interviews. Then comes the 1 hour group interview in which about half of the interviewees are selected.
After years of reading applications and interviewing applicants, I find that it isn't as important whether a person has already done research/other volunteering activities/or has a very impressive resume. What I use to select applicants is that they show effective communication skills in their essays and during their interview. During the interview, I look for applicants that show respect to others as they are speaking (since it is a group interview), respect to me as the interviewer, and who is genuinely interested in volunteering in the hospital and is not just looking for the experience to be a "resume filler." I think that if you show a good personality, a desire to help others, and a genuine reason for applying to the position, you will be able to get the position despite of a lack of experience. After all, these are the only traits that are needed for a good volunteer at a hospital. If you don't get selected the first time around, apply again. I always give a preference to applicants who have applied more than one time, because it shows their dedication.
Lastly, like some of the above posts, I would suggest that if you still cannot obtain this position, look for a smaller hospital or a clinic. I volunteer in a small clinic myself in which the entrance was not competitive at all, but I find my experience in the small clinic to be just as rewarding, if not more rewarding, than my experience at the large, competitive hospital.