How is an Uno stalemate handled?
According to the UNO rules, drawing without playing is a legal move:


  RENEGING
  You may choose not to play a playable card from your hand. If so, you must draw a card from the DRAW pile. If playable, that card can be played, but you may not play a card from your hand after the draw.


However, the intent of this rule is more likely to allow a player to strategically hang on to a card (for instance saving a wild card for later) than for someone to hold onto all the cards.

Given that the goal of the game is to get rid of your cards, in the normal manner of play this is a scenario that should never happen, so the rules do not cover this case. 

A few possible options for a house rule for when this happens:


If a player can't play and can't draw, their turn is skipped and the next player goes
The game is a draw
If you're including the scoring rule, the game ends, and you each score points for the cards in your opponents hands