I come to you with a vocabulary question and a terrible illustration.
I am looking for something much like a pogo stick, but a great deal shorter, perhaps four to eight inches. You could also consider it a shock absorber, but I thought those were hydraulic. It would need fasteners on each end, plus some mechanism to keep it together.
They'd come with a variety of spring constants so I could Hooke's Law out how far it would move under a given force.
Does this thing exist, and if so, what is it called?
I guess that'd just be called a spring. A spring and a shock absorber are two fundamentally different things. A spring reacts with a force proportional to how it is compressed or tensed. A shock absorber reacts with a force proportional to the speed with wich its end is displaced.
Why do you need to encase the spring in these cylinders?
Depending on the forces involved you could use pvc pipe or perhaps get a telescopic fork from a crashed motorcycle. When bent the fork tubes are basically scrap metal, but they're usually bent within the top 6 inches or so. If you cut out a section where the two pipes have a sliding joint you'd have what you want and it would bear great radial loads too.
A shock absorber is a combination of a spring and a dashpot, you could get one with an external spring and drill a hole in the dashpot to drain the oil.
What are you trying to do with it? You can get pneumatic cylinders with springs in them which are very similar to what you are looking for.
Wouldn't it be easier to have a separate structure constrain whatever motion to that one axis and the spring do the spring part? That way you could just buy some rods and sliders like they use to build homemade CNC mills.
If you absolutely have to, maybe look into gas springs (think cars and cupboards)? AFAIK they're not built to bear any significant radial load, but we still have no idea what you're building.