Here we go again! Songs do not have to only possess chords that are diatonic!! Who spreads this misinformation?!
Playing an F chord after shows that the key would indeed be F major, not that that is ever reached - but it sounds like home more than any other.
So, yes, it's ii and iii of key F, possibly attributable to being in G Dorian. Which of course, is made up from the parent key F major.
Just dug out a version, written in key C, but that ends on the aforementioned F! It's by Bacharach and David, so will be a 'good song' anyway! Just taking that line doesn't really help find a key - far more of the chord sequence would be needed, and it's not available in the version quoted.