I didn't think I'd started work long enough ago for anything to be strange but actually I have remembered some from reading the other responses!
Defo remember being told to do a CV on nice paper, with colours (subtle) and fonts. Can't quite get my head around making it boring and easy to read /print /photocopy now which probably doesn't help 
I worked in a shop selling cigarettes and you could buy tobacco products from age 16 but you had to be 18 to buy a lighter because of the solvent. Had several arguments about this. I think 16 year olds were allowed to buy matches, though. We were just allowed to eyeball people's age and only ask for ID if they looked underage, rather than under 30. That tended to be a bit dodgy if you were young yourself and didn't know how to judge people's ages very well, and anyone over 20 would often get shirty about it, so challenge 30 probably isn't a bad policy as at least it's now expected.
Vaguely remember cheque guarantee cards and not being able to process a return/refund for anything paid for by cheque until 3 working days later in order to give it chance to clear. When we moved to chip and pin we stopped taking cheques at all, which greatly annoyed some people.
We had a special machine with a database of every single book in it which we could use to order books in (whsmith) as long as the customer knew the ISBN, title or author, but it was a separate terminal to the normal tills, right at the back of the shop so if you were the only one on the till you had to phone downstairs to get someone to come up and do the order.
Not long after I started, our till system was upgraded and I remember being fascinated that it became a mini computer running on Windows with a tiny but full keyboard. Totally normal now and any non computer epos system would be seen as antiquated.
Paper gift vouchers that never expired, and you'd give change out in vouchers up to the whole £1 and up to 99p in cash. No useless gift card with 32 pence remaining.
I worked for a graphics studio for my first full time job. We provided content for screens in waiting rooms, which was very high tech at the time! I would make up the graphics and string them all together, and then they would run through a dedicated computer which would overlay it with hold style muzak and run it through a TV which was attached to a video and DVD recorder. About half the surgeries (doctors and vets) ran their waiting room info screens on video, rather than DVD. There were also no adverts! They had the practice info with staff, opening times, phone number etc and the rest of the time was filled up with information taken from NHS leaflets or health campaigns. The vet ones might have info about their yearly subscription service for vaccinations etc but that was as close as you got to an ad.
Because it was run through a TV to be recorded even for the DVD versions, it was totally fine for the graphics to be a bit "spiky" or not smoothed out and you had to be careful with certain colours and delicate details which would be lost in this rendering process.