I was born and raised in Oregon, and unless it's changed, the minimum for elk and sheep is .243, for deer, lions, and bear its .22 CF. I have never shot an elk with my 6mm(same caliber as .243), but I used it on deer with handloaded 95gr nosler partition and got excellent penetration and great results. I'm sure the 243 could and has worked countless times on elk, but I would suggest something with a heavier bullet. If recoil is a big factor, the 7x57 or 6.5x55 will give practically the same recoil as a 243 but with a much more appropriate bullet.
But hey, like someone's avatar says (I can't remember who to give credit to), this post was worth exactly what you paid for it.