We worm-compost nearly all the vegetable scraps, eggshells, and the advertising fliers (torn up in strips) that come in the mail, it's amazing how much the volume is reduced (slightly disappointing actually, using it to add to the garden soil.) We only avoid composting citrus (due to the toxicity to a lot of invertebrates) and really tough stuff (fruit pits, avocado peels.) Lately we've had some fly problems, so we've been throwing fruit scraps in the trash instead, it seems to be helping.
I've previously composted in a Rubbermaid tub that was about 8 inches deep which I had drilled lots of holes into and placed on a couple boards to keep it off the ground. It worked, but far better, in my experience are semi-continuous worm composting systems where you put food and scraps of paper in the top, and remove finished compost from the bottom.
There are two types of these for sale that we've used, and you could probably build some variant of either one of them: One is a series of plastic trays which nest in each other and just have an open mesh as the bottom of each tray. You fill up a layer (it's nice to have about 5 of them), and once you've filled them all, and hopefully the compost on the bottom is nearly done, you take it out from under, leave it on top of the stack uncovered, stir it a bit to expose the worms to light and dry the compost out a bit, at which point the worms retreat (mostly). The whole stack sits atop a tray which helps catch the runoff from watering the compost.
The other type, and I think it's possibly the best due to greater aeration (and probably not too hard to make) is a tapering fabric bag that you hang from a PVC pipe frame. The top is some kind of mesh fabric with a zipper around the perimeter, the bottom has a drawstring closure. The body of the bag is some kind of heavy synthetic cloth. You want a bucket under the bottom to catch excess water and periodically you can open the drawstring and squeeze / bang on the bag / scoop some compost out the bottom.