I have an ~120 litre aquarium containing 30-40 small fish (tetras and barbs).
What volume of plants is enough to keep it in balance? I'd prefer not to introduce new entities to the water for as long as possible, aside from feeding the fish.
Tetras, in general, also like some plants to hide them from the light.
Since there are so many specimens of plants, with so many shapes, sizes, densities... I think the best way to "measure" how much plant you could have would have to take into account all these characteristics.
What I would recommend would be high, floating plants, covering 30% of the surface of the aquarium. And planted plants to cover about 20% to 30% of the area of the front of the aquarium.
Well, "in balance" could mean a lot of things, but I'm guessing you mean you want to add plants to keep your water quality in check. If that's the case, the better solution might be to look for improvements in your filtration and maintenance routine first: 30 or 40 fish is a pretty full load for a ~120L/30gal tank, so there might be more effective, or even necessary, things you can do without adding any new organisms.
You won't be able to completely eliminate the need for basic maintenance like water changes or siphoning just by adding plants. Even tanks that are heavily planted with few animals can't escape that. Plants do help reduce some waste products like nitrates, but there are other things that will get "unbalanced" over time, like water hardness and solid wastes. The plants themselves are a source of maintenance too, since they all have different lighting, nutrient, and substrate needs that your tank may or may not meet as-is. (A deep discussion on that would be off-topic for Pets, but there are some excellent resources out there, like The Planted Tank, APC, and the Barr Report forum.)
That said: they do help keep your water quality stable, make your tank look better, and your fish more comfortable. They're a valuable addition to a fish-oriented tank. So again, make sure your filtration and maintenance are enough for your bioload as a first step, because you'll need to do that with or without plants. Then do some research to find out what luck other people have had with your particular lighting, substrate, and fish community. Most plants will grow and reproduce if they do well, so you can start with a few to get a handle on them and then add more later.