It is safe enough to store hardy squashes in an apartment, but more delicate ones such as zucchini and other thin-skinned squashes should probably be stored in the crisper of a refrigerator if you're planning to keep them for more than a week. If that space is not available, you can extend the "keep" time of less hardy squashes with just a bit of advance prep before storing. Seal the stem and blossom ends of the squash with melted candle wax, beeswax or paraffin. In a very clean sink, create a mixture of sterilizing water by making a strong brine, then adding one tablespoon (yes, tablespoon; it is, after all, a sterilizing agent) of bleach per gallon. Sink each squash in this rinse for a few seconds (I generally don't do it for more than 3 seconds, 5 if the peel is uneven or has scarring) and pat excess moisture off with a clean, non-abrasive, absorbent cloth. Allow the exterior of the squash to air dry before storing per the method for hardier squash.
For hardier squash, find the coolest location in your apartment that doesn't include a damp place like a bathroom or laundry. If it has a closet, you just found your storage place. Put down either a few small boards or a few bricks with several layers of cardboard. Perfect base. You can store your squashes right on there, or put them into cardboard boxes on top of that base, which is what I usually do. Don't let the squashes touch each other, as every point of contact is a potential weak spot. Turn your squash once a week; handle very gently to prevent even shallow, minor cosmetic bruising or abrasion of the skin, again in the interest of avoiding weak points. If you live in a relatively humid climate, melt candle wax or paraffin onto the squash where the stem meets the fruit -- after ensuring it is completely dry, of course -- to minimize the formation of molds and fungi.
Check hardy squashes for soft spots or mold each time you turn them, and immediately remove — and use — ones showing those signs. More delicate squashes should be checked every three days and should not be stored longer than two weeks, in case there is spoilage from the inside.