The progression begins with Am, the relative minor chord in C. The next chord is D, which is outside the key of C. It’s a secondary dominant, the V chord in the key of G. In European classical music, this chord would typically lead to G, which would then resolve to C. However, this is not what Kalinnikov does. Instead, he follows D with Fm. This is another chord from outside the key; it’s borrowed from parallel C minor. Kalinnikov set you up to expect a move toward the sharp side of the circle of fifths, but instead he goes several steps over toward the flat side. The last chord is the tonic C. Kalinnikov gives the chords some extra flavor by repeating the notes E and C over all four of them. These notes are chord tones in Am and C, but they create colorful extensions on D and Fm, turning them into D9 and Fm(maj7) respectively.
I naturally found the title very intriguing and started arranging something, using that chord sequence, earlier today. Will record my ideas tonight. Didn’t listen to the samples and will wait to hear them until I am done with my recording using the same chords. I don’t even know the tempo or how quickly the chords change.
If you didn’t know, melodies are copyrighted but not chord changes. Thousands (millions?) of songs use the same chord changes.
Oh, and by themselves the chords don’t sound very sad to me. But I may have a high tolerance for sad music. 😄