You forgot about the water heater :)
IMHO, a forty year old furnace, and its companion, a forty year old water heater, are best left out for the recycling scavengers. Some people will try to repair them, and some will even be successful, but to me, it's not worth it. They both tend to rust out in critical areas, and become unsafe if not properly serviced by someone with good experience.
Buy new for those ones, again, in my opinion.
The refer (refrigerator) can be repaired in many cases, since what usually goes is the sealed system of tubing containing an ammonia-based liquid that tends to rust internally with age and eventually the rust can/will block enough of the internal passages it goes through to impede the flow. This results in poor or no chilling effect.
The good news is that you can buy a replacement 'rebuilt' sealed system for many of the refers out there. The bad news is they seem to cost $500 or more.
Most AC units don't have good parts availabilities, one of the exceptions to that is Armstrong. People seem to be able to find off the shelf parts to rebuild and even improve on the design of them in one case I read.
But really, if your budget is stretched really tight, you can buy alternate (and relatively cheap) heaters for your trailer to replace the furnace function, and some folks will put a house refer into the trailer, if they don't travel often.
A complete set of appliance could easily run $4,000 for new, so I can't blame people that look for alternatives.
If it's to be, it's up to me.