No, there is no single fixed constant time. It depends.
Pretty much all instruments allow the player to vary the time for which notes are held. In some cases very extreme (a Hammond organ allows you to trivially hold notes indefinitely, but also allows millisecond-short staccato), in some case only subtly (a xylophone or bass drum doesn't have much sustain to begin with, but with active damping can you make the sound yet a bit shorter).
A sound that only lasts for 1⁄10000 s is indeed pretty useless, it certainly won't have any distinguishable pitch – basically, it'll just sound like a very sharp snap/clap. But a sound that lasts for multiple seconds is also not often very useful, because it allows hardly any musical movement without everything ending up as just a cacophonic jumble.
Even what range of durations is desirable depends highly on what musical setting the tones are to be used in. For very sombre or æthereal music, there may be only few notes most of which last a second or upwards, whereas virtuousic pieces may have countless very short notes of mostly only tens of milliseconds each, which normally gives a pretty restless, energetic result. (Clever use of such short notes may give you an end result that's again rather dreamily-textured though.)
If I were to give any concrete bounds on what note lengths should be considered, I'd say 30 ms − 10 s, but really I don't think it makes much sense to use that in any rigid way. It is possible to start music with a fixed grid of, say, 32th notes of 60 ms each, but that's pretty limiting – it makes anything that's not pretty hectic more complicated than necessary, whilst on the other hand still not allowing truely sharp staccato.