I would've said Biomedical Science because there is very little to no patient contact however its still not for the faint hearted to be honest.
Radiology or certain fields of research might not be a bad idea.
radiology is a branch of medicine, and requires a medical degree, hence lots of exposure to blood & ill people. and interventional readiologists can do various things which involve blood.
I work in a pharmacy and there are people who go in with cuts/wounds asking the pharmacist to "have a look" if they don't think it deserves a trip to A&E. Also people come in asking for their dressings to be changed, and parents bringing their children in if they've fell and cut their knees/heads/elbows outside. You may not see much blood as a pharmacist, but it's unrealistic to say you won't see any.
Sell prescription drugs illegally.
If you would be all right with taking bloods then many things would still be open to you.
Why not look into something like dermatology or general practitioner. Most medical professions involve taking blood as it is a key part in diagnosing many conditions but if you are fine with that, those two fields are pretty much blood free (apart from the tests) but still very 'medicaley'
I think it's really hard to avoid blood ON patients in healthcare. You could be on a rehab ward and still get a fall with blood everywhere and head injury, or iv access falling out, coughing it up, vomiting it up, pooing it out, weeing it out, or wounds dehissing. Some professions it may be rare, but it could still happen and be unavoidable!