Asuming 50% voter turnout in every state, what is the minimum amount of votes to win the presidency?
I don’t know the answer, but I think I can tell you how to determine it. Because electoral votes are equal to the number of Representatives plus Senators and because Representatives are approximately proportional to population, a candidate earns more electoral votes per each popular vote by winning the small states. So the minimum number of votes required to win the Presidency would be to pick the smallest states until you’ve accumulated 270 Electoral votes. You’d then need just over half of the voters in each of those states and none in the other states.
You’d have to modify this a little bit in case the smallest states in order didn’t sum to exactly 270 Electoral votes. So you might want to pick a slightly larger state in place of one a bit smaller to just get there. It’s also possible in Maine and Nebraska to split the Electoral Vote.
But if you want to get technical about it, the answer is zero. What you need is to have The Electoral votes split 269-268-1 amongst three candidates, with a “faithless” Elector casting his vote for someone who received no popular votes. Since no candidate has a majority of Electoral votes, the House decides from amongst the top three recipients of Electoral votes. And they then choose the guy with the single vote and no popular votes. Yes this is incredibly unrealistic, but so is winning half the Electoral votes and not getting any votes in the largest states.
I ran the numbers. To get exactly the 270 votes you need to be elected, you’d need to carry the following states: Virginia, New Jersey, Washington, Arizona,
Indiana, Tennessee, Missouri, Maryland, Wisconsin, Minnesota, Alabama, Colorado, Louisiana, Kentucky, South Carolina, Connecticut, Iowa, Oklahoma, Oregon, Arkansas, Kansas, Mississippi, New Mexico, Nevada, Utah, West Virginia, Nebraska, Hawaii, Idaho, Maine, Rhode Island, New Hampshire, Wyoming, Alaska, Delaware, Montana, South Dakota, North Dakota, and Vermont plus the District Of Columbia. These states collectively have a population of 134,351,932, of whom 67,175,966 voted (per the 50% rule in the OP). We’ll assume you got exactly 50% plus one of the votes in each state for a total of 33,588,023. Your opponent got 33,587,943 in these states.
Your opponent also got all the votes in the remaining eleven states: California, Florida, Georgia, Illinois, Michigan, New York, North Carolina, Ohio, Pennsylvania, Texas, and Massachusetts. These states have a combined population of 172,654,618. With the 50% rule, that means he received 86,327,309 votes in these states plus the 33,587,943 he got above, for a total of 119,915,252 votes.
So the results are:
You: 33,588,023 votes - 270 electoral votes
Him: 119,915,252 votes - 268 electoral votes
Congratulations, Mr President. You now have a mandate from the people.
Your numbers are too high, Nemo. A 50% voter turnout means half the registered voters voted. You used the total population rather than the number of registered voters.
The abolute numbers may be different, but the ratio between the two vote totals would be the same, assuming the math is right and all states have approximately the same percentages of registered voters.
If so, it’s theoretically possible to win with less than 25% of the popular vote.
Wait wait, in US when you measure voter turnout you only look at the people who actually register to vote, not at the the total number of citizen who are legally allowed to vote in the first place? 
Or, for that matter, “zero – followed by a majority of ‘faithless’ electors.”
So really the minimum is 219 votes.
(1 faithless elector + 218 votes in the House)
In order to win a state’s electoral votes, you don’t need to win a majority of votes cast, but only a plurality. To give just one example, Bill Clinton won all of Colorado’s electors in 1992 though he won less than 40% of the popular vote in the three-way Bush-Clinton-Perot race. In a ten-way race, you could theoretically win all of a given state’s electoral votes with just over 10% of the popular votes.
Every state currently holds popular elections to decide the allocation of electoral votes, but it’s not a constitutional requirement. Any state could change its rules to allow the state legislature to appoint electors. That’s the way electors were appointed in most states in the early years of the republic. George Washington got few, if any, popular votes.
I’m leaving that as an exercise to the reader.
So really, the answer is 22 - two votes going to you in the following states (and everyone else in the state voting for themselves):
CA, TX, NY, IL, FL, OH, PA, MI, GA, NC, and NJ for 271 Electoral College votes.
Wait, wouldn’t this be the right answer? Electoral college votes are split 269 / 268 / 1, vote goes to the house. The house (435 voting members) splits 146 / 145 / 144, with the 146 House votes going to the candidate that had one electoral vote. The new President would have had 147 individuals voting for her! (Unless the House has to choose amongst the top three electoral college winners by a majority, not a plurality).
Is the right answer 147? Do I win a prize?
P.S. I think my answer, if my assumption is true that the House of Representatives is able to choose the president via a plurality, has a smaller number of votes than Munch’s answer. I would argue that in Munch’s case, the president would have garnered 22 votes (popular) plus a minimum of 270 votes (electoral) for a total of 292. You get a smaller total if you go with the one faithless elector, then a person can become a viable candidate for the House election with just one vote.
But only a quorum (50%+) is required to do business in the House. So, if you had 218 members present (and the other 217 with more important things to do than decide the President of the United States) the guy could be elected with 110 (109+1) votes in the house, plus the faithless elector=111 votes total.
A win for me in the thread?
But wait! I just checked the 12th Amendment. I was wrong, the House needs a majority to vote for the president. It says
I am now changing my answer.
To elect the president, you need a quroum in the house. The quorum would be a member (or members) from 2/3 of the states. 2/3 of the states is 33 or 34 states. So 34 people, each representing a state, could be a quorum for electing the president in the House, right? Then it says “a majority of all the states shall be necessary to a choice”, meaning that 26 states have to choose the same person. (I suppose the hypothetical could have more than 34 representatives present, as long as there are at least 26 states represented by only one person, and those 26 people end up being the “majority of all the states”, whatever that means exactly.)
So my answer changes to:
1 faithless elector picks X, rest of electoral college is split 269/268 between Y and Z.
House votes on X, Y and Z. They have a quorum small enough that 26 people represent the “majority of all the states”. A majority of the states (26 representatives) vote for X.
X became president with 27 votes (one elector, 26 representatives.)
Am I right or am I right? Right?
I’m changing my answer: in this case, each state would only get one vote.
So, each state has one representative vote for a candidate while the other representatives abstain. 26 states vote for the winner, 24 for the loser.
26 reps + 1 faithless elector = 27 total votes for the new Prez.
Is my amended answer right?
CURSES CURSES! 
You beat me!
Or, I amend my answer again. Only the full delegations of 34 states are present.
One votes for a candidate, while the others abstain.
Winner 18, Loser 16
18 reps + 1 faithless elector = 19 total votes!!!
WIN?
Except the amendment says you need a majority of all the states - wouldn’t that mean more than 50 / 2 “yes” votes? If they mean a majority of the states present and making up the quorum, then I think you are the winner, jtgain.
I’m not sure if you need the full delegation of the state to represent the state - maybe one representative for the state (with the other ones not showing up) is enough.