What happens when I try to exile more cards from my library than there is in my library?
You don't lose the game until you try to draw a card that doesn't exist.


  104.3c If a player is required to draw more cards than are left in their library, they draw the remaining cards and then lose the game the next time a player would receive priority. (This is a state-based action. See rule 704.)


Since Golos exiles cards it doesn't lose you the game, but that will probably happen on your next draw step anyway.

You can now exile your 2 cards to Golos's ability:


  609.3. If an effect attempts to do something impossible, it does only as much as possible.