Apples are best stored at 0° C (32° F) at 90% humidity. If you don't have a cellar, you'll have to try to recreate that environment as best you can.
The closest you can come in most small apartments is actually a plastic bag with some holes poked in for ventilation (to prevent excess moisture from building up). Then put that in the refrigerator to get close to the correct temperature; the best location is the vegetable crisper where you have control over the humidity. Don't overpack them, though - about 80-90% full is the densest you should go.
Also be sure to pick only the best apples you can find if you plan to store them long-term. Over-ripe apples will obviously not last as long, and any bruising or piercing will speed up the decay significantly.
Edit: If it's a large number of apples then you can store them in a crate, which provides adequate ventilation, but good luck getting that into the refrigerator. I would still separate them into plastic bags, put whatever I could in the crisper, the rest in the fridge "proper", and put out any that I intend to consume within a week or two in a regular fruit bowl out in the open. If that's still not enough, then you probably need to accept that small apartments aren't very well-suited to storing very large amounts of fresh food; just because they're easy to get, doesn't mean it makes economic sense.