To be classified as a "Western film," (especially an Old Wild Western film, as best can be done) there must be straight, Historical Factors involved. Then, it becomes more realistic, because the stories would be real as it really happened at that time during the 1800's.
My personal opinion is that the classic western has evolved into modern westerns. The classic westerns were usually set in the western U.S. in the 1800s, and they had certain cliché elements which were not all required, but some combination of them was expected. These included Indians (usually portrayed unsympathetically), cowboys, settlers, sheriffs, bandits, land barons, saloons, shootouts, covered wagons, the saloon girl with the heart of gold, and many more. But there have also been modern westerns made and set in modern times, and I would include The Misfits as one of these, and also Bad Day at Black Rock. They're both set in the west. The Misfits has cowboys in it but makes the point that the cowboy way of life is disappearing. Bad Day at Black Rock has no cowboys, but it does have an isolated western town barely reachable by train, and a hero who arrives and saves the town from the bad guy.
A lot of people have commented that Star Wars is a western set in space (and it even includes a "saloon scene.") I would call Star Wars a sci-fi film that refers to westerns, but not a western itself. However, I don't think westerns necessarily have to be set in the west if enough other elements are present. For example, Drums Along the Mohawk seems like a western to me, and it includes an Indian raid, but it was set around the time of the American Revolution and in New York State, which was the frontier at that time.
I've participated in similar debates about what makes a screwball comedy. I don't think there's any single element that has to be present in a western, it's more the general gestalt, and the same with screwball comedies (although the latter DO have to be at least a comedy!).
This is a question to which there really isn't a wrong answer, and I love the answers given thus far. When the 1992 version of The Last of the Mohicans came out, Madeline Stowe stated it was a Western, due to the fact that that was the farthest West anyone had been on the continent. This being the case, I finally understood why people classify Drums Along the Mohawk as a Western.
I personally feel that a Western is a film set in the West c. 1865-1900. Anything afterwards is a Modern Day Western. It doesn't have to have action, per se. (Four Faces West is a perfect example.)