I've got to go with the crowd on this one. Pretty much any of the popular calibers will do the job at 300 yards. But here is the trick, you need to shoot some targets at 300 yards (and everything in between) and know your gun. All bullets are going to have some drop to them and you need to know how much at various yardages. And this is when the optics on your gun will be important. I would recommend something with at least x12 magnification. A 3x9x40 will work but a 4x12x40 will work better at 300 yards.
And don't get caught up in looking at ballistic charts so that you can get the flattest shooting bullet you can find. There are many fine flat shooting bullets but you need to get the bullet you want and then learn that bullet. For me personally I've always done better with heavy slow bullets, but I know many people that have done great with fast light bullets. You have to figure out what works for you so don't even worry about how much drop. A bullet with 3 feet of drop at 300 yards might put more deer in the freezer for you than a bullet with 1 foot of drop?
I think this also brings the oldest and most debated way of thinking regarding hunting cartridges. Do you go light and fast or heavy and slow? Both approaches work, you just have to figure out which one fits you and your personality.