With the right bullet in a hot load OK; though honestly, I'd prefer at least a hot .270 (140gr at 3100fps is a max load, vs 2950-3000 for the same weight from a .308, or about 100 fps lower for either with a 150gr; and it has better sectional density, and better ballistic coefficient with most bullets), or a 7mm mag as a minimum.
The .308 and .30-06 are good for velocity, and great for bullet choice; but worse on penetration in the higher velocity loads, because of their comparatively low sectional density.
Some of the hot 6.5s (the long european, or the short magnums) would be a good choice as well, in the heavier bullet weights (at least at shorter ranges). I don't think I'd go for anything below 6.5mm though. A mid weight or heavy 6.5 gives you a great BC, and a great SD.
UPDATE: A commenter notes that the .30-40 Krag was considered an excellent cartridge for elk up until World War Two (technically til the late 30s I'd say, when high velocity chamberings started becoming popular). Again, I note this is because of sectional density.
The standard .30-40 hunting load was 220gr at about 2200fps, up to a max of about 2400fps; considerably heavier than the standard 150-180gr hunting loads for .308 and .30-06, in the same diameter (though also considerably slower. A typical 180gr load from an '06 is around 2700fps). That's sectional density.
Not to say that I'd take .30-40 over .30-06; for one thing, you can shoot '06 near as heavy, and with a significantly higher velocity; but it does mean that yes, in the right load and at the right range (a short one, because the .30-40 bullets tended to have pretty poor BC) it would take an elk cleanly.
Sectional density is also why the light weight hypervelocity magnums aren't really great for elk. They just don't have the mass or sectional density to penetrate deeply on a consistent basis, on an animal with that much muscle and that deep a chest. They are near perfect for smaller deer species however.
Sectional density means greater penetration for a given energy level. Remember, you can take Elk with a bow an arrow, and the right shot placement, because that arrow is going to penetrate deeply with all that weight behind a small (and sharp) diameter. THAT is sectional density.
Hit the retained energy floor necessary to penetrate deeply (10"-14" is ideal), with the mass, and sectional density necessary; and you're going to take the game cleanly.
...now if there was only a consistent way of figuring out how much energy and sectional density were actually necessary.
