Can I shoot an eagle?

If you mean hawk, yes you can. You have to be within range, but if you do shoot one, you'll be able to collect the hawk feathers and hawk beak for alchemy as they're fairly hard to obtain.

Is it possible to hit a bird? Yes.

Is it incredibly difficult?

YES.

The best place to try is Solitude - you can use the bridges between the towers to get to a decent altitude, the birds fly relatively low, and you're in a separate zone from the main game world, so range is less likely to be an issue.

Having a full complement of Archery perks makes an enormous difference as well, for obvious reasons.

With a full compliment of archery, you can use the right click of the mouse to zoom. Your sightvisor doesn't move everywhere. It's really easy to kill an eagle in Solitude, Eagles which are flying over the port.