Not possible biologically? I'm not entirely sure if that is correct. Yes, there's some aspects that wouldn't really fit, such as living for an eternity and living off of blood only (not very healthy). But some attributes to a vampire does make sense. Here's a list of some of them that I can think of:
1. Sunlight can kill them-there is such thing as being allergic to light, although it is rare, and if memory serves right, in extreme cases, this allergy can kill the victim if they stay out in light too long, forcing them to come out only at night
2. Becoming a vampire after being bit-I have read that some people associate vampires with rabies, which transfers itself through a bite, so it is logical that a vampire is really caused by a virus and is transfered through a bite
3. Developing fangs-there are surgical methods to getting fangs artificially implanted as well as a disorder that does cause development of fangs, called NEMO, although this disease is fatal and sufferers never live to be more than a few years old, however it doesn't seem farfetched for there to be other diseases that cause malformation of the teeth
4. Sensitivity to garlic-another rabies trait, sufferers of this disease are very sensitive to smell and garlic would drive them crazy
5. No reflection-Again a rabies trait, if memory serves right, those suffering from rabies cannot look at themselves in the mirror, so, in a sense, they "have no reflection".
6. Consumption of blood-while it is not possible for humans to have a diet made entirely of blood, blood consumption by humans is nothing new, sucking a cut finger, for example, is instinct that both humans and animals have (wound licking), and there are cultures who mix blood with their food and others who do drink blood as part of a ritual.
If I made a mistake with any of my explainations, I apologize. This is all from memory.
As far as whether or not I believe vampires exist, I'd have to say yes I do, but they are not the monstrous near-immortal vampires of legend or of the undead. Just people either suffering from a disease or are self-proclaimed vampires who drink blood for the sake of rituals.