Inside cleaner in a car wash.
At my college work-study job in the library, one of my tasks was to insert cards into the card catalogue.
My first job (in 1975) was as a switchboard operator for an answering service. The old fashioned “one ringy-dingy” kind with jacks that plugged into sockets that corresponded to phone numbers for different busnesses. Totally obsolete, of course.
Six years later I joined the Navy and was a computer operator. Our systems used magnetic tapes, data drums and punch cards – all obsolete now. In addition to the actual computers, we used specialized machines to manage the punch cards, to sort them and collate them and so forth – these machines are also completely obsolete.
I was a bagger at Meijer(very large grocery store chain in Michigan).
I have seen that all baggers are gone from there now, requiring the cashier to do it for the customer.
Having said this, I assume other stores still have baggers, but not my job at Meijer. 
I had a job where I would go to government public disclosure offices and get massive amounts of documents. The I would enter than information into databases that would be sold to companies. No doubt that information is currently available on the web for free.
I worked as a video store clerk. That is one it’s way out.
Paper boy. If it’s not obsolete now, it will be soon.
With the gradual implementation of self-checkout lanes, the grocery cashier will eventually be phased out as well.
I worked for a financial consultant firm and used to have to manually call Municipal Bonds and Mutual Funds and find out what the going rate for the various accounts - that’s all electronically pulled down now.
I used to be a page in a library - I could easily see large libraries at universities or such places implementing robots that could re-shelve books themselves, with only a token human to deal with issues such as jerks who would turn the books the other way on the shelf to confuse the robot.
I read about one of the big businesses that’s all warehouses - Amazon or something - having robots that automatically grab merchandise from the warehouse, a library version can’t be too far away.
I worked as a credit clerk. If a customer wanted to charge something more than the “store limit,” the cashier would call me. I’d look up the customer’s account in a giant paper ledger. If the customer’s recent history checked out, I could okay the purchase. If not, (charging above the limit, multiple purchases at different stores in a short period of time, etc.) I had to call in a supervisor. My duties also included calling the stores one at a time if a customer reported a card lost or stolen and something we oldsters called “filing” – which involved taking paper forms and putting them in a folder with the customer’s name on it, then placing those folders in alphabetical order into large cabinets.
These days, almost all of that is done by a computer, and instead of having to summon a supervisor maybe 10% of the time, it happens maybe one time in a thousand.
When I was a kid we used to ride our bikes over to the bowling alley on Saturday mornings to keep score for the adult leagues. We’d keep score for three games and come away with 10 or 15 bucks, all of which would be immediately changed in quarters for deposit into arcade games and pinball machines. Of course bowling scores are kept by computer now.
Speaking of obsolete jobs in bowling alleys, my dad worked as a pinsetter.
Pump jockey for a Texaco station.
See? Told you you could always trust me. 
My wife and I were in New York before Christmas and visited Tiffany and Company on 5th. They still have elevator operators in fancy uniforms. As you said, there can’t be many places that do.
It’s not obsolete in Oregon. You’re not allowed to pump your own gas here.
I’ve had several photo-developing related jobs that are either obsolete or nearly so now. One was at a one-hour photo place downtown that served mostly real estate agents and government offices that documented needed city repairs (inside big sewer pipes, etc.). Those folks all use digital now, and I believe the developing machine I used is probably rare – I think most places, like supermarkets and corner drugstores, that offer real film developing and photo printing send the work out.
One summer I cleaned out and cataloged the contents of a university darkroom used for remote sensing imagery (mostly photos taken from aircraft of forests). Not only is that all done digitally these days, I should have had eighteen kinds of safety equipment and probably some kind of license to handle some of those chemicals (not all of it was just your standard color developing; I’m not sure what it all was).
One year in college I was the photographer for the theater department, taking publicity shots for the local papers and the school catalog, etc. I developed the B&W film and prints myself by hand in the college darkroom. I don’t think they have a darkroom anymore – I think photography students have to rent space at a place in the city. And the theater people went digital.
Before all that, I worked in a small public library, and of course one of my duties was making cards for the catalog for new books, which is all gone now. I also reshelved books; I know they still do that by hand (usually by the hand of ill-paid high school students like I was).
I’m only 40!
Not me but my husband spent the first years of his adult working life as a commercial photographic re-toucher in New York. He retouched advertisements and catalogs and stuff like that. Now it’s all Photoshop.
I used to do “paste-up” at the magazine where I worked. We’d print out the articles from Word (or a typewriter, going even further back a few more years) and send them off to be typeset into neat little columns. We’d get these strips of paper back from the compositor and would have to trim them with X-acto knives and then run them through a wax machine to apply a wax coating to the back. Then we’d carefully place them on the “boards” (big sheets of heavy paper with grids printed on them to aid in lining things up) and send these laid out mock ups off to the printer. We also had to use “border tape,” which was tape with lines printed on it, to create the borders around the articles or pictures or whatever. Had to lay it down nice and straight, cut the ends with our handy-dandy X-acto knives, and if we put it down a tiny bit crooked… then it would end up printed in the magazine crooked. I learned this skill at the student newspaper in college.
All of that is now done digitally in InDesign or Quark (and back in the day, PageMaker). Completely useless obsolete skill.
- Pershing Missile Crewmember – during the mid-1980’s I worked with Pershing 1A missiles in Germany. They were eliminated by a disarmament treaty.
- Trade entry clerk – worked at the Chicago Mercantile Exchange and would take the hand-written cards from the traders and runners and enter them into the computer. Now it is all done on handheld systems.
- Computer operator/tape librarian – maintained the magnetic tapes and reels, mounting them for job runs and doing weekly backups.
- Y2K testing and system verification – ‘nuff said.
- Long-distance phone number verification – Before you had the ability to dial 411 and get a phone number from anywhere in the country, you had to dial the area code plus 555-1212. I worked at a company that was combining the numbers from all of the baby Bells across the country into a single database that they would sell to the phone companies. We had 160 million residential numbers and 60 million business listings. I oversaw a team whose job was to track down reports of incorrect numbers and fix them in our system. Now the phone companies have their own databases.
- Ant killer – my wife is no long afraid of them so she doesn’t need me to kill them for her anymore. Fortunately she’s still scared of spiders, wasps and other creepy crawlies.
Obviously the job that I lost a year ago when our company closed… sigh
I got computers ready for Y2K. I spent endless hours going through endless programs, correcting the ones that had only 2 digits for the year.
Not too much call for that these days.