Many doctors and others in the medical profession still carry beepers and pagers.
Lawyers and those in the legal field prefer WordPerfect to Microsoft Word.
Urban planners are wedded to old-school listserv mailing lists.
What other professions seem to embrace technology that may seem oddly archaic, compared to the practice of their peers in other professions?
Acoustic guitar players.
Embedded systems engineers love rs-232.
And buggy drivers, and Model T mechanics, and VCR repairmen, and Amish furniture makers, and Amish barn builders, and Amish buggy makers, and Amish anything.
Really, I’m thinking about mainstream professions that hold on to some archaic or obsolete technology when a modern substitute is available, not those whose work depends solely on that archaic technology, and without that archaic technology they would not be able to practice their profession. For instance … oh, meteorologists prefer weather radar displays that use CRTs rather than LCD displays. I just made that up, but you get the point.
Writers who still use typewriters.
I think we had a thread on why docs still use pagers and it was determined that, compared to cell phones, they’re cheaper for the hospital and less prone to dead spots.
When I was in the Army, I worked on the M47 Dragon, which was designed in the 1960s. I finally stopped working on them when Fort Riley got rid of their stock in 2003.
We also drove “deuce and a half” 2 1/2 ton trucks, which have been in service since the 50s.
I’m sure military folks could come up with list several pages long with little effort. Needless to say, the military uses and maintains its equipment as long as it possibly can.
And most doctors have a Cell phone/blackberry/Iphone/whatever personal hand held device- it’s just the Hospital’s aren’t going to PAY for that stuff usually.
Hospitals want to save $ and pagers are cheap and effective, but plenty of doctors keep phones on them as well.
CRTs are still necessary in many vision sciences because LCDs have temporal or spatial weaknesses which may affect any experiments.
Drgon was developed in the 70s, I believe.
The B-52 Stratofortress, a plane that has been in service for over half a century, is the king of long-lasting modern military technology. Well, it shares that honor with the deuce-and-a-half, really. Still I’m not sure that’s in the spirit of the OP. The B-52 is still a first rate airplane; it was just ahead of its time in the 50s, that’s all.
If you think about it, one of the most archaic things the armed services hold on to isn’t a technology… it’s the concept of drill. Drill’s purpose was to control troops in the field in a time when exact movements were necessary to maintain effective battle formations. It’s of no practical value now, and yet it was like half my basic training, or felt like it.
Really? I know a couple older profs who do - but, well, they’re old. 
And old people routinely hold onto older tech. In my (very limited) experience, I haven’t seen a government or nonprofit office running on wordperfect.
BZZZT wrong! But thanks for playing. 
both of these are mere children compared to a Ma Deuce. Been around since WW I.
Professional chess players
Chefs (excepting food processors and microwaves)
I know a lot of professors who highly prefer pen and paper to typing.
NASCAR racers still use carburetors.
The M2 Browning .50 cal machine gun has been around since 1923 and is still in use today. Sometimes there is just nothing to improve on.
A minority of professional woodworkers use hand tools only. There are probably very few who actually make their living using strictly hand tools (many prepare their wood using electric planers and jointers and then use hand tools for the joinery), but there are definitely some. There are always debates and discussions (sometimes heated) on woodworking messageboards about hand tool versus electric, and use of the term ‘hand made’ to promote one’s work. It’s a pretty touchy subject with woodworkers.
And I’m not talking Amish at all here. Just regular woodworkers, usually working in small, proprietor-owned shops. Almost all of them could use the latest in electric equipment, but some choose to stick with hand tools instead.
Okay - how about electric guitar players who use tube amps - old tech when solid state amps are readily available (but, IMHO, can’t do what tube amps can(link to old thread about this)
Up until about 10 years ago ( when I left the business, may still be true ) theatre professionals had a inordinate fondness for analog clocks backstage and consistently rejected digital clocks in favor of these.
Huh? This is like calling someone old fashioned for driving a Chevy instead of a Ford.
Militaries worldwide are still using firearms technology that hasn’t progressed in quite some time. Metallic cartridges date to the mid-19th century. Smokeless powder to the late 19th century. Belt-fed machine guns are only slightly newer than smokeless powder. The “assault rifles” issued by virtually all nations are all variations on an idea and technology dating to the 1930’s.