Far from everyone governed Athens. Adult male citizens had a vote. Adults were men who had at least begun their military service. Citizens had to be descended from citizens - after the reforms of Pericles? from both parents, excluding children of Athenian men and 'foreign' women (450 B.C.). Non-citizen resident males had no vote. There was almost no possibility of movement from non-citizen to citizen status. Estimates of the numbers of citizens at any time in Athens are speculative.
Like other vote-based societies in the ancient world, one had to be physically present in order to vote. Military service or simple distance prevented the exercise of citizenship. Voting took place in public, sometimes by physical division ("Everybody for Plan A go the right....") and sometimes by written ballot. Ostracism was only by written ballot (a name scratched on a potsherd or ostracon).
A number of positions of the Athenian Democracy were filled by lot or random choice of a citizen from a pre-determined group. For instance, the chief operating officer {that's the best I can do on the fly} for the day, the Chairman of the Prytany or Council of 50 was chosen by lot from the 50. Having served once that man could never serve again in his life.
See also: