Ikea has some flat white bins that have a diagonal side that hinges open. We bought one and put it under our usual garbage can, and put the bottles in there. Without spending the entire day looking through catalogues (the italian one is different from the american one so the products may be slightly different) i found this, which is similar to the one we got. Ours has the opening on the narrow side, and there's enough room for a big kitchen garbage can on top. They;re intended to stack and to be able to open while stacked, so you can put stuff in them without taking one off the other; I didn't look through to see if there were other models on the american catalog.
Also i always squash my plastic bottles so they take up a lot less room (small apartment). Takes a second.