Airgun hunting is all about accuracy. Even a very powerful airgun is pretty wimpy compared to a very weak firearm.
You can hunt very small animals (mice/small birds) with your airgun out to the distance that you can consistently make center of body hits. Larger animals (up to about squirrel-sized) can be hunted out to the distance that you can consistently make brain shots.
A body hit on a squirrel-sized animal with an 500fps .177 airgun is pretty unlikely to result in anything but a wounded animal escaping.
In general, I hate this kind of question because the truth is that hunting with an airgun requires a lot of shooting skill. People who ask about hunting with airguns rarely have the shooting skill to consider hunting with them.
Ok, I sorta assumed that you were talking about airgun hunting even though you didn't specifically ask the question. Here's a more general answer.
The trajectory on a 500fps .177 airgun is going to have a very pronounced arch. I recommend that you get a large sheet of paper and draw a large dot near the top of the sheet. Then hang it with a safe backstop where you have some distance to play with. Shoot 5 shots at 5 yards and then move back 5 yards and repeat. Keep doing that until you get to 50 yards if you have that much distance.
Now go back and look at the groups you've shot. First of all, if there's any wind, the longer range groups aren't going to look much like groups. Second, past about 20 to 25 yards you'll notice that the drop is becoming very pronounced. It doesn't take much distance until you're dropping enough that making a ranging error of just 1 yard will guarantee a miss.
Add to that the fact that airgun pellets are not very aerodynamic and that by 50 yards, that pellet's probably only going around 300fps and you rapidly realize that even if you can make the hit, it's not much of a "hit" if you get my drift.
One last bit of information--most folks agree that a conventional airgun shooting conventional airgun pellets is safe past about 400 yards. So unless you have 400 yards of range to play with, you need to be careful about having a good backstop (one that will consistently stop a pellet) and avoiding ricochets.
Have fun shooting! I learned to shoot with airguns and I still enjoy them a great deal.
John
Disclaimer: My comments about airgun power are applicable to conventional pump-up and spring-piston airguns. Pre-charged pneumatics can be a good bit more powerful and approach firearm power levels in some cases.