So much depends on the scope of emergency you are planning for.
If it's the collapse of civilization kind of thing (Power grid gone and not to be restored, breakdown of the monetary and food distribution systems) then an apartment or even the 'burbs would more or less be a deathtrap (Of course, you could personally safeguard anything in it, since you wouldn't be going to work yourself with no gas and no functioning economy, but that would mean you'd better have about three years worth of MREs in there with you, a nearby water source, and the will and ability to forcefully defend your stuff.
If it's a voluntary evacuation situation and civil order hasn't broken down, and so probably temporary, and you'd have the time, mobility, and freedom of movement to retrieve your gear, a storage area would work fine, not so good if the authorities go into a mandatory evacuation or lockdown approach though.
These are the two ends of a spectrum, but as you can see, only under the sunniest scenarios would it work to have stuff you would really need be more than walking distance from your living space.
