Can I shoot an eagle?

Solution 1:

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.

Solution 2:

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.

Solution 3:

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.

Solution 4:

go to the archery menu and get the zoom with the bow ability and the extra damage and range. As well as get the one where when you hold your breath you can slow down time to get a better aim, if your stamina is high. You can hold down your breath for a solid 30+ seconds. also you can earn a shout which nearly freezes time. The 2 together can make them really slow. Solitude is a good place. But you can also use lightning.