Is it possible in UNO to run out of cards to draw?
Yes.  Here's how:

10-player game.  Designate players 5 and 10 "lucky", meaning that these players monopolize the number that initially flops, and also monopolize the wilds, ensuring that they always manage to play, and the other eight "unlucky" players are never able to play.  Assign two of the four colors to players 1-5, and the other two colors to players 6-10, ensuring that the current color is always inconvenient when an unlucky player's turn comes around.

Let's say players 1-5 got the colors blue+green, and the red 1 initially flopped. The "lucky" players proceed to play out the other 7 1's, with the unlucky players drawing (8*4=32) cards in the process, leaving only 5 cards in the discard pile.  Player 10 plays a Wild, calling one of his own colors, and then the endgame begins:


Player 1 draws a junk card
Player 2 draws a Wild Draw 4 and plays it, calling one of player 10's colors
Player 3 draws the last 3 cards from the deck, including a Wild Draw 4, then reshuffles the discard into the draw and also draws a Wild.
Player 4 draws a previously played blue or green 1
Player 5 plays a Wild, calling one of his own colors
Player 6 draws a previously played red or yellow 1
Player 7 draws a previously played red or yellow 1
Player 8 draws a previously played red or yellow 1
Player 9 draws a previously played red or yellow 1
Player 10 plays a Wild Draw 4 that he has held since the start
Player 1 draws draws the last three cards from the deck (all 1s), then reshuffles and draws a Wild
Player 2 draws and plays a Wild Draw 4, calling one of player 10's colors
Player 3 reshuffles that Wild Draw 4 into the draw pile and draws it, -there are no cards left to draw-
Player 4 has nothing to play and no cards to draw


Here's a sample deck order for producing this result: 
https://pastebin.com/hzBiFQ0B