I just figured out in this tune the chords F major and e minor fit, and they are repeated all over. Surely its a simple and repetitive piece, but what I am wondering is, the only key where both chords fit is C major. So is this in C major? But then nowhere a C major chord is played, is this normal? I somehow have the feeling a C major should be played to establish it as a C major key, but maybe I am wrong. Could someone explain? And if it is C major, then does it happen often that music is written in some key, but the "tonic chord" is not played?
Maybe you could say that the key is Phrygian E minor. The key signature for Phrygian E minor would be just like C major and A minor, no sharps and no flats.
The part of the track you linked to has a strong minor feeling and when the bass note change from E to F the F feels like a leading tone down to E, and it does go down from F to E again and again.
Note that in the two chords E minor and FMaj7 the tone E is "persistent" in that it is there all the time at the top of the chords while the base goes E-F-E-F-E etc and the chords follow the bas line of course, but the top is an E.
All in all it gives a strong feeling of E as a tonic, thus you can argue that the key is Phrygian E minor.
EDIT: Just like @Michael Curtis I am only relating to that part of the track where those two chords are alternating again and again. I listened to that part but did not listen to the whole track.
The first thing to clear up is music doesn't need to be in a key.
In this case it may be better to say
F both fit into the key signature of 0 sharps/flats.
You could call this C major, but why not call it A minor? Both chords fit into either "key."
Em F C: iii IV Am: v VI
The dominant chord really defines a key. (You could alternately say the leading tone defines the key.) Nevertheless, it would be unusual to not have the tonic chord as you have pointed out.
Interestingly with this case, neither the tonic nor the leading tone for either C major or A minor are used. Any claim that it is in a key would be weak unless additional musical context defined a key.
For practical purposes of notation you would use a key signature of 0 sharps/flats for this music.
I notice that your YouTube link drops into the middle of a long Chillwave recording. I didn't try to figure out the chords from the recording. I only answered regarding the two chords you listed in your question. But, you may want to consider a larger selection of that track when considering what key it might be in. The harmonic pace in Chillwave is often slow and repetitive so a larger selection may be needed to get the complete harmonic picture.
I can hear either E minor or F major as the tonic chord in that music. But it doesn't really matter. Functional harmony isn't the game this piece is playing.
With the major chord being Fmaj7, the second one could easily be Cmaj7, with an E bass. Em and Cmaj7 are easily confused for each other. If that's the case, then it's similar to Erik Satie's Gymnopedie.
Having said all that, it's music that's painting a picture, a mood scene, and there's no compunction for it to have a particular key base, just a pallette of sounds which interact with each other; no-ones going to be whistling or humming it, although I suspect it's actually C major.