How to store items safely?

MOST containers are perfectly safe. All containers you make with the workshop ARE safe.

However, I wouldn't store items for long in instanced areas just in case. The Quarry area in particular, I lost all the items I stored there during my siege upon the place after the water drains out. Something about that place resets, so beware and don't store stuff there. So, if an instance has different quest tiers especially, watch out.

I have noticed that container items respawn with a high frequency in places I've already raided. Not sure why this is. Something to do with either continuing after a death instead of loading up a save.

Early Game. It looks like the workbench etc, share storage in one area and link between settlements later. However, I have had some pre-built containers reset with new items. I didn't save anything that way so nothing was lost thankfully. However, if you build your own containers, of any type, in a settlement area where you could expand, those containers are safe.
For instance, at Red Rocket Truck Stop you have everything except the Chem Lab and the one at Sanctuary is just up the road. I cleared out some space left side of the Power Armor Station (by recycling what was there) and made a bunch of storage units for weapons and outside next to the Armor station I made a bunch more. Those have been fine as far as I can tell.
Careful though, some don't allow you to stack another of its type very well. Play with it a bit. You can always recycle if you screw up (just make sure what you recycle is empty). I threw in a bed later because the Sanctuary settlers were always grabbing all the beds when I wanted a quick heal.