Two possibilities:
1) The kind of composters that stay in one place, but you can spin them (like this one here :
https://www.compostumbler.com/StoreFront/IAFDispatcher (there are less hardcore looking ones as well!).
There are some like this that divert the finished compost to the outside barrel, while the unfinished compost still works.
2) a worm composter (the worms eat the garbage and leave 'worm casings'--the PC term for worm poop, LOL!-which is excellent for the soil. You can also keep these composters in the house).
A third type is a double-decker type (I can't figure out what they are really called), which allows you to remove finished compost while the new compost is still working.
Unless you plan to fill the composter, then let the entire batch 'cook' til it's done, it's really best to have either two composters (which is what I did because I already had one) or a composter like the double-decker or the tumbler that separates done from undone.
Another (quite cheap) option is to buy some chicken wire, make two tubes with it (that you fasten to the ground with ground staples) and throw your leaves and food scraps in them. They work especially well for leaves.
I have two composters as well as a chicken-wire leaf composting area right now : a plain square one and a round rolling one (that acts like the tumbler, but you have to roll it around the yard). I had a worm composter in the past, but I didn't keep up with it--it's not a lot of work, but too much for lazy me. If I ever have to replace my composters, I'd go with the tumbler type.
HTH!
ETA: I didn't read other responses before I wrote mine, but I agree that if you don't have a lot, and you have a corner to put to this use, a pile is just as good as a formal container. My stepmother does this (and she and my dad started doing it in the 60s!). She just keeps a bucket in the kitchen, fills it with compostables, and dumps it daily.