While I doubt the gun safe will be an issue, I personally prefer to keep my seeds in the back of the fridge, in a container with desiccant. I currently have a couple of hundred of them in an old vitamin container with a couple of packs similar to yours. Individual strains are in labeled straws that I pierced with a straight pin to allow the desiccant to work inside the straw as well.
I say this, because in nature, seeds typically drop in the fall, go into dormancy through the cold months, and sprout as the weather begins to warm. Keeping them refrigerated will mimic nature more closely and thus, they seem to come out of dormancy a bit faster. Just my personal experience.
Granted, I have never had issues when I didn't follow these rules until the seeds were several years old.
