Removing the drives in no way will do any damage to the rest of your parts; assuming you power down the computer first (and I unplug it from power as well for safe measure).
The antistatic bags are a good "just-in-case" if you have any, if you are storing in them in a room with hard surface (wood, tile, vinyl, etc) then it wont really be much of an issue.
SSD is very durable and will survive most any condition you should have in a home.
HDD is a little more delicate. Dont store it near any semi powerfull magnet (magnetic screw driver or metal screws is not a problem), and store it where it wont fall (or in something protective).