This would be a frightening situation. 270towin is a web site that provides a map where you can guess which states will be won by Hillary Clinton and Donald Trump. By accident I entered the states shown here as won by the two candidates. The result came out to 269 for each. The scenario I suggest is not beyond logic.
What happens if there is a tie? Congress will make the choice. The Congress meets in joint session on January 6, 2017 to count the electoral votes (this count happens whether the election is close or not). If no candidate has reached 270 Electoral Votes, then the House and Senate take over and elect the President and Vice-President, respectively. Note that the newly elected Congress will be sworn in on January 3, 2017. It is that new Congress that takes on this responsibility. If each states gets one vote the winner would be Donald Trump as more states have a majority of Republican congressional districts in the current session and that is unlikely to change.