Omen Machine and no library
You would not lose the game from being unable to draw a card.

This is because the rule that causes you to lose the game states (empahasis mine):


  104.3c: If a player is required to draw more cards than are left in his or her library, he or she draws the remaining cards, and then loses the game the next time a player would receive priority. (This is a state-based action. See rule 704.)


and Omen Machine says (emphasis mine):


  Players can't draw cards.
  At the beginning of each player's draw step, that player exiles the top card of his or her library. If it's a land card, the player puts it onto the battlefield. Otherwise, the player casts it without paying its mana cost if able.


So what happens is:


Omen Machine attempts to exile the top card of your library.
You don't have a top card to your library, so the following steps don't do anything.
You continue onto your First Main as normal, having not drawn a card and not exiled a card. 


This is further supported in the Gatherer Rulings of a similar card, Maralen of the Mornsong which states:


  While Maralen is on the battlefield, no player can lose the game due to being instructed to draw a card with an empty library.