It's different.
My big fear was heights, when I accidentally became a techie I overcame that because I had to - quite simply, if I didn't dangle off the top of a ladder, no-one else would do it for me. I think that similar things happen to people when faced with the need to overcome that fear.
As far as medicine goes, although I'm alright with blood, there are plenty of other things (particularly big, gaping, rotting holes in people) that really make me wince. But that's alright so long as you can get past that. In your case, it's alright to be a bit urgh when taking blood so long as you can still do the job while being urgh.
The other factors are, first, not letting the team down and second, not loosing face. They're both pretty big motivators if you have the right attitude towards medicine.
In short; you'll be fine.