I have shot plenty of Red Stags which are a bit smaller than Elk and found them easy to kill. A .243 would do OK. I have also shot plenty of Sambar deer which are between Red deer and Elk in size, and found them hard to kill. A 338 is a good choice. I saw plenty of Elk while on a Mule deer hunt in New Mexico and was impressed by their size, but have always wondered, really, how tough are Elk to kill in comparison to other animals?