"hunting caliber"...
30-06 and 300 win mag guys hunt with, I guess you could use a .338. But at the cost of a .338 is a major factor that will keep practicing down to a minimum.
If you asked me what a hunting caliber was, I would have to ask what you are hunting for. 223, 22-250, 243, 270, 308, 30-06... these are for anything small to elk etc.
Guys do shoot .303, but I don't think there are any modern firearms made in that caliber anymore.
As Robert Nicholson mentioned.. the pic you posted is of military full metal jacket, non hunting type ammunition.