I'm just going to agree with all these points:
1. check the guitar setup. Put a capo on fret 1 and see if the guitar (as a whole) is easier to play - ie does basic fretting need less pressure. If it does, the nut needs lowering. (It should feel as easy as that with no capo.)
2. practice the shape further up the neck. It shouldn't be much easier to press the strings down (on a well set-up guitar), but the frets are closer, which helps. This is not just to get the F working, but that barre shape is the standard one for all major chords not available as open position chords, and also for movable versions of those open chords. I.e., eventually you will be using that shape all over the neck, not just for F.
3. play the middle 4 strings with one finger each (ring 5, pinky 4, middle 3, index 2). Use the thumb to mute the 6th string, and lean the index over to mute the 1st string. That's a perfectly usable (and complete) F major chord, and quite easy to change to and from a C chord (the usual companion to an F).
If you can lean the index enough to fret the 1st string as well the 2nd (double-jointing) - and/or wrap the thumb to fret the 6th - so much the better! But that's not necessary if you can mute those strings.
4. Other options for F (not necessarily practical if other chords in the song are in open position):
x-x-3-2-1-1. Many beginner books suggest barring that 1st fret (good if you can do it), but you can play this with 4 different fingers (pinky-ring-index-middle). Obviously it's hard to mute the 6th and 5th strings if you're strumming this shape - which is why it's not really practical in the long run. (The middle 4 strings shape is much more usable.)
x-8-7-5-6-(5). The top string note is optional, and can be muted by the index if you can't get the index to barre the top 3. Again, for strumming, you can mute the 6th with the thumb.
x-8-10-10-10-(8). Again the top string is optional, so this is doable without a barre - unless you want to use a ring finger barre on 10. If you can do the index barre, then you can take it all the way across to 6th... but then if you can do this barre, the other 6-string barre shape shouldn't be a problem...
