If there’s one thing that Republicans and Democrats can and do agree on, it’s outrageous election rules that make any viable third party practically impossible.
This presumes a demand for a third party, which I see little evidence of. Further, while the dominant parties have rigged some of the rules to favor themselves, those obstacles are quite minor and could easily be overcome by any organization popular enough to become viable in the first place.
The main structural reason for a two-party system was put in place by the Framers before the advent of political parties in the U.S.: a single member district electoral system. Since Congressional districts are winner-take-all, political parties or candidates who appeal to a narrow ideological segment aren’t going to fare too well. There’s no prize for second place. Thus, with SMD, as night follows day, comes catch-all parties: moderate, bland parties that can appeal to something close to a majority. The alternative is some version of proportional representation, but that is often chaotic and has weird consequences of its own. In PR, it’s often the case that the tail wags the dog: a fringe party gains disproportionate control because it’s the key to a coalition government.
The US Senate was initially an appointed office, although it, too, was at-large and thus had to gain a consensus in the state legislature. It’s been an elective office since 1912 and, again, SMD.
The presidency, as we saw so clearly in 2000, is elected in similar fashion: 48 of the 50 states plus the District of Columbia award all their electoral votes to the candidate who garners the most votes statewide. (Maine and Nebraska both grant two votes to the statewide winner and one to the winner in each of the Congressional Districts). Third party candidates have won state electoral delegations as recently as 1968. But, again, the down side is exemplified by Ralph Nader, who was strong enough a candidate to garner a substantial number of votes but not nearly enough to win a single state. But his running split vote of the center-left coalition enough in a close race to cost Gore the election.
A candidate who isn’t a Democrat or a Republican can win in any of these races, but they have to capture the public imagination. See Jesse Ventura or even Lowell Weicker. There were no structural barriers to Ross Perot winning the presidency in 1992; he simply alienated too much of the electorate to have a chance. And, once a candidate starts to fall, the pressure to not “waste your vote” comes into play, as we now see in the California recall contest.