Raw meats can be picked up by hand. They are near impossible to cut safely when using gloves. Most contamination of raw meats comes from equipment and not hands. Bone in saws are the worst offender at contaminating meat (so if you want it to last longer, avoid bone in cuts or find out which cuts were done by hand or on a boneless saw). Also, go early in the day and buy things that were cut the same day (saws can be operated for I believe up to 4 hours of use before requiring cleaning). Every cut at a given store has probably been touched by hand at least twice before it gets in front of a customer.
Ready to eat meats (deli meats, ham, cooked chickens) can only be touched in a glove and have to be served at below 38 or above 160ish degrees(can't remember exactly what it is).
I worked in a meat store/deli for a few years in high school/undergrad. Some of the best hungover work days of my life were spent behind a deli counter or in front of a band saw. At least I made it out with all of my fingers. Good times.