A president can win an election with only 129 electoral votes and the reason for that is that if no presidential nominee gets 270 electoral votes the House of Representatives decides the winner in a vote. In this vote each state gets 1 vote each regardless of how many electoral votes they have. So if the Green Party wins 26 states they will have 26 out of 50 votes which means they will win the vote in the House of Representatives. If these 26 states are the states with the least amount of electoral votes the Green Party wins the presidency with 129 electoral votes.
There is also a way to win with 124 electoral votes which is a little bit more complicated. Nebraska and Maine have implemented the Congressional District Method which means that the party who wins the whole state gets two electoral votes in both Maine and Nebraska. But then there are 2 electoral votes up for grab in Maine and 3 in Nebraska. To get these votes you need to win in the Congressional Districts and Nebraska has 3 Congressional Districts and Maine has 2. This means that if the presidential nominee wins the state but loses in each congressional district he/she will only get 2 electoral votes in Nebraska and 2 in Maine. If this happens the presidential nominee will get five fewer electoral votes than in the first example but the nominee still has a majority in 26 states. This means that it is possible to become president with only 124 electoral votes. Calculation example in Nebraska.
1st District: Green Party: 49,000 - Red Party: 51,000
2nd District: Green Party: 49,000 - Blue Party 51,000
3rd District: Green party 49,000 - Red Party 51,000
Total votes in Nebraska:
1: Green Party 147,000
2: Red Party 102,000
3: Blue Party 51,000
The Green Party wins the state but the Red and blue Parties share the Congressional Districts.
Sources:
wikipedia US Presidential Election
wikipedia Contingent Election