What's the best setting for drama, awkwardness, and first-person online video games? Yes, an elevator. If you don't have a real elevator available, why not play Elevator: Source, a "co-op elevator experience" in which a group of players hang out in an elevator as it travels between floors in a surreal building, stopping periodically to show players what's on the floor outside the elevator. Some floors contain kittens in space, others are infested with dinosaurs, and still others do unexpected things like drop you down the elevator shaft. The future of elevator gaming is now!
The game features dedicated buttons for checking your watch and coughing. While I haven't played it, I watched a good chunk of this 45-minute gameplay video. It's kind of amazing -- the game perfectly captures the strange pacing of enforced elevator waiting, the awkwardness of unwanted passengers (a man with a shotgun gets on at the first stop and nervously twitches for a few floors), and the banality of listening to lame music as you wait.
Note: this gameplay video contains NSFW language -- it's a group of dudes playing a multiplayer game, you can expect a fair bit of swearing and crudeness. Skip ahead to 1:45 for the beginning of the actual elevator journey.
If you want to actually play the game, check out this site -- you'll need Half-Life 2: Episode 2, some mods, and various other prerequisites. The video is probably enough to give you an idea of what gameplay is like.