Election Day 2008. The votes come in, and are pretty much the same as 2004, except Obama has picked up Nevada, New Mexico and Iowa, which went Republican in 2004. That leaves Obama with 269 electoral votes, and McCain with... 269 electoral votes.
Uh oh, the Presidential vote goes to the House for the first time since 1824. Let's say there are no too-radical changes in the composition of the House. There can be some sweeping out of a few marginal seats of the minority party from their states, maybe, but only a few key races. For instance, Lynn Jenkins (R) beats out incumbent Nancy Boyda (D) in Kansas, and Travis Childers (D) loses the rematch with Greg Davis in Mississippi.
So, the vote goes by states, with the states voting in the order they were admitted to the union. The winning vote from each state's delegation is then counted as a single vote for that state. Let's say everyone votes on party lines. Everything goes as expected through the first 38 votes. After Colorado's Democratic delegation (the 38th) votes for Obama, Obama leads 22-16. Obama is certain to win Hawaii and Washington state when they get their chances to vote, and the next vote is Earl Pomeroy (D) who gets to decide the fate of North Dakota's vote on his own. Obama lost the popular vote in North Dakota, but not by much, and Pomeroy is loyal to his party, and so he votes for Obama (23-16). This sets off a political firestorm, and there is huge pressure for the next one to vote, Stephanie Herseth Sandlin, who likewise single-handedly controls South Dakota's vote. Fearful of her re-election chances, she votes for McCain, bringing the total to 23-17. As expected, Montana, then Washington bring the vote to 24-18. Then a run of hard-core Republican states (Idaho, Wyoming, Utah, Oklahoma, New Mexico) brings it to 24-23 and to Arizona, where the delegation is deadlocked in a 4-4 tie. Finally, they agree that with McCain having won the state's electoral votes, and it being his home state, that has to be the tiebreaker, and it's all evened up at 24-24. Alaska and then Hawaii bring it to 25-25 and another deadlock.
So then the Senate votes for vice-president. Everyone votes along party lines, except where they've endorsed differently. Lieberman and Chafee cancel each other out, and Biden wins over Palin easily. But no president has yet been elected still. The House has to vote again. This time there is HUGE pressure on Pomeroy to change, but also on another "odd man out" representative: Mike Castle (R) of Delaware. Now, he has an incentive to change his vote, and does in this second round. But Pomeroy changes as well, and the vote ends up right where it was. 25-25.
And so it goes, on through two weeks of deadlocked 25-25 votes... And on January 20th, 2009 there is the historic inauguration of President Biden.