It is possible, but can be very difficult (depending on the fingering... more on that below). Most pianists would prefer their right hand, simply because most are right-handed.
I was messing around with it a little bit and thought of a few different fingerings. For reference, 1=thumb, 2=index, 3=middle, etc. The most intuitive (but probably not the easiest) way would be 1 and 4 on the C and F, 2 and 3 on the D and E. It's very hard to keep the fingers in sync this way, though, and the difficulty increases with the tempo. Though I am sure there are virtuosi out there for whom this synchronization is not an issue, I am not one of them.
But since the D and E are adjacent white keys, you could actually hit them both with one finger. This takes care of the problem of keeping the inner notes in sync with one another, but what of the outer notes? I ran into trouble when using 2 for the inner notes, because I could not find a fingering that made it easy to keep the outer notes in sync. The same happened when I tried using 3 for the inner notes.
Then I made a pleasant discovery: if I use 1 for the inner notes, I can comfortably use 2 and 5 for the C and F, respectively, and I have no problem keeping the notes in sync. In fact, with this fingering, I really have no problem keeping these trills going cleanly and clearly for basically as long as I want at maximum tempo.
So to recap, here is my recommended fingering: 2 on the C, 5 on the F, and 1 on the D and E together. Piece of cake!
P.S. All I have right now is a little electric keyboard without weighted keys. Until I try it on a real piano, I can't tell you with 100% certainty that this fingering is as good as I think it is. I'm going to say I'm 85% sure at this point. Hope this helps!