How long do I have to train to be able to do 1000 pushups?
Impossible? I think not:
"The world record for the most number of non-stop push ups is 10,507 by Minoru Yoshida of Japan, which was achieved in October 1980, breaking the record of 7,650 by Henry C. Marshal (USA) from 1977. Some time after this record, Guinness stopped recording non-stop push up records (no rest periods allowed) and instead started recording most push-ups in 24 hours with as many rest periods as they wanted."
As to how long you'd have to train to achieve such a feat would depend on any number of factors like your size and build, and most importantly the ratio of fast-twitch to slow-twitch muscle fiber in your body.
Doing very large numbers of repetitions of anything requires endurance and at this level it has essentially become an aerobic activity.
If your body has a high ratio of slow-twitch muscle fiber, and your general build is fairly small, then you're going to do better at such extreme endurance activities than a large, heavy-muscled person with a lot of fast-twitch muscle fiber.
(Such a person could likely lift far more weight in a max bench press, however. "Strength" and "Endurance" are quite different.