The fastest that a man has been able to run till now is a distance of 100 m in 9.58 seconds; this was achieved by Usain Bolt in 2009. His average speed here was 37.57 km/h.
To be able to run 1000 m in one minute the required speed would be 60 km/h. It is highly unlikely that a human being can achieve a speed of 60 km/h and maintain it over a duration of 60 seconds to cover 1000 m for some time to come.
Though it wouldn't be right to say that this can never be done; with the constant advances in science and technology many physical feats that were considered impossible in the past have been performed; the one being referred to in the question may just be something that is on its way to be achieved.