Well, in a way, I've done that. I left my second tank fallow for 6-months thinking that it would eventually work the nutrients out of the system through the nitrogen cycle. It was teaming with pod life after just a few weeks. It just ran on it's own with lighting and the rare water change.
However, if you mean supporting fish, corals, etc., I doubt it. Just look at the nitrogen cycle. It releases Nitrogen back to the atmosphere, so what is fixing nitrogen to maintain that cycle without constant feedings.
In the wild, plankton do a lot of work, but maintaining that in a microcosm would take a large amount of volume. In theory, it's possible, but I would think it would be unstable for anything but the largest setups.