Sodium persulfate is a strong oxidizer, which is what makes it an effective etchant, but that also makes it a fire risk. In storage, keep it cool and dry; silica gel moisture scavenger packets probably aren't a bad idea. And, of course, keep it away from anything flammable.
Finally, it's a good idea not to dispose of the used etchant solution down a drain. Check with your municipality to find a safe disposal method, rather than add it to the ecosystem where it will poison wildlife.
If it were me, instead of a sealed container I'd just run the process under an open window with a fan propped in it to exhaust whatever fumes develop, and maybe under an improvised fume hood made of a cut-open garbage bag and masking tape. (If you do that, make sure to leave space at the bottom for intake air. The idea is to make a funnel that will pull room air in, mix it with process gas, and exhaust the lot outside instead of into your living space.)
The window-and-fan method worked well when using vinegar-based iron acetate stain, and later polyurethane sealant, on a quite large table I made back when I lived in a tiny apartment; for a low-volume process like this, it'll probably do just fine, I'd think.
I might try a sealed container with some kind of valve or loose seal and see, thank you!