I've recently moved onto the second major hub area in Deus Ex: Human Revolution. So far, I haven't come across anyplace as safe as my apartment in Detroit. I don't feel like lugging a rocket launcher with me everywhere - is there a safe place I can stow extra gear, or can I drop it anywhere and it'll be there when I return later?
- City hubs with sidequests are temporarily safe to "store" things in, until you leave the city hub, and the game always warns you when you do this about how side-quests will be un-completable.
- No other mission area is safe. Sometimes the game tells you you won't be able to return to an area, but don't count on it. If nothing else, there are some one-way elevator rides, etc.
I left a fully upgraded silenced pistol on the floor of my apartment in front of the "stash" area and when I returned, it was nowhere to be found (much to my intense rage). Luckily I had an old save file that still had it. My recommendation is to leave ammo and health and non-essentials there, or to just sell them. Any weapon you really like should probably stay in your inventory, or make sure you save a file before you leave it on the floor.
I'm not sure your apartment in Detroit is safer than any other non-restricted area. If anything, your apartment is a worse place to store items since its out of your way most of the time.
I'm not 100% sure if they will be remain when moving from city to city.
IMO the best bet is the Hengsha Whore Hotel, somewhere near the weapons dealer, available from the very start of first visit in Hengsha; I think that only storing some spares makes sense here, as Detroit is visited more times and covers more of the gameplay. Also, slight story spoiler,
don't place the items in Alice Garden Pods - that area will get off-limit for/at some time.
To sum up: simply use the streets; no NPC will bother with your items at all, and they will stay there until the riots start.
After that, you're up to your arm augs to help you manage your stuff, as you won't be getting back to Detroit after that, and your following visit in Hengsha will also be the last.
Note that some subareas do get reset with time (doors re-closed, barrel/boxes/computer terminals reset etc), but the loot doesn't.
Source: my recent gameplay;
all the items left in various places during the very first visit were still there during the riots