Storing a compass near un-magnetized iron/steel should not be an issue. Magnets work by having all there atoms spinning in the same orientation (simplified model). Two things can scramble this uniformity, either physically striking it hard enough to knock all the atoms out of alignment or by introducing a conflicting magnetic field.
The conflicting magnetic field can come from either a permanent magnet or simply from an electrical charge passing through wires. A weak extrenal magnetic field from an electrical cord can be as damaging as strong magnetic field from a speaker, it just will take the weak field longer to de-align the compass atoms' spins.
Non-magnetized iron/steel will attract the compass's needle, but all the compasss atoms' spins will stay aligned with each other. However, depending on how the metal was worked/treated some of its atom's spins may have ended up aligned. I happen to have a compass on my desk. I've pulled out my Leatherman Kick and pointed it's pliers at the compass, causing the needle to spin around and point it's rear towards the LM. Switching the LM around causes the other end of the needle to swing towards the LM. This indicates that my LM does have some residual magnetism in it. I can't pick up a paperclip with it, but there's enough there to radically affect the compass.
Now I'm doing the same with a peice of N80-steel pipe. The compass needle spins to point at this peice of steel, but reorienting the steel in regards to the compass does not change the direction of the needle. In fact if I pull the steel away and approach the compass from "behind" the rear of the needle now points towards the same spot on the metal. This peice of pipe has no residual magnetism in it.
I could lay the compass on the steel for years without any damage/disorientation of the compass atoms'spin alignment because there is nothing "pushing" at their spins. If I set it on my LM Kick the magnetic field from it will push at the spins and slowly cause them to "wobble". Different atoms in the compass will feel different amounts of push due to both distance from the pushing field and inhomogenaity in the magnetic field. The end result will be a mishmash of spin directions in the compass needle and so it's magnetic field will no longer be strong enough to line up with the Earth's magnetic field.
Now if I set the compass on a strong permanent magnet something slightly different can occur. In an undamaged compass all the spins align parallel to the needle. If you leave the compass next to a strong magnet this external magnet can realign all the spins towards it. The spins will still all be aligned, just not parallel to the needle. Now the north and south poles of the needle aren't at the tips but rather somewhere else along the needle and these new "poles" will point north/south.
If you want I can go on and talk about why the production technique of the pipe removed any possible magnetism while making the LM Kick may have slightly magnetised it. If you really get me going I might start adding vector diagrams and stuff like that.
-Blast, on a Friday afternoon