For Monument Valley, it depends o what your notion of "possible" includes and how much time you want to spend stopping and seeing things.

It's a six-and-a-half hour trip. one way.

So it would be a very exhausting day of driving, imo, to do it in a one-day out-and-back.

And you wouldn't have much time to stop at some of the fascinating sites like the John Wesley Powell Museum, or Navajo Bridge. If I were doing it, I'd plan a stop in Kayenta (a very nice Hampton Inn there - the only one I know of with its own restaurant). Plus, the Amigo Cafe is always worth a stop for lunch or dinner) in one direction or the other.