I have fired at paper as long as 1,000 yards away with .30-06 and .308 Win out of rifles that were <1 MOA shooters at 200 yds. I've even hit what I was shooting at, though no "X"s were harmed in the experiment.
The longest hunting shot I would take is <300 yards...I've hunted strictly for meat so I keep within distance I know I can hit within a 6-9" vital zone. All 6 of my hunting rifles (a .25-06, a 6.5x55 SE, 2 .308s, 2 .30-06s) shoot to <1 MOA at 200 yards...from a rest...while I am seated. As I do use shooting sticks or a bipod when hunting, if possible, or a decent field position seated or off my pack, frequently after a bit of a hike, sometimes uphill, in the cold with heavy clothing, I find my additional likely error is 1-2 MOA on top of the <1 MOA of the rifle. At 300 yards, that's a 9" circular error from my POA...about a deer's vital area. An elk's vitals might be 15-18", but I want reliable vital area hits, so even my elk shots will stay within 300 yards.
Now I actually practice from field positions using the equipment I'll take with me and, while I consider my self a pretty fair shot, to quote Dirty Harry, "a man's got to know his limitations." While I know, and have practiced with my rifles at various ranges with and without holdovers (I have RapidZ reticles on 4 of my rifles and duplexes on 2), I challenge most hunters (myself included) to distinguish 1-2" differential holdover at 275 to 300 yards...and that's the difference between the MBPRs of the .30-06 and the 7mm Rem Mag. I also find that the two rounds out of the same rifles (I've tried the 7mm magnum out of the M70 EW, the rifle I frequently use in .30-06) present me about the same felt recoil with 150 gr bullets.
Net, net, either caliber is a perfect match for hunting almost anything in North America, though if I was limited to Brown bear Country in the lower 48 or Alaska, I'd probably opt for either a .338 or .375 mag...not for distance but for hitting power. I have considerable personal experience with Rem, Win, Savage, and Ruger rifles and recommend all of them (as they have worked well for me). Pick either caliber, though the .30-06 will be marginally more flexible and cheaper because of the choice of components and weight of powder. Practice a lot, and think long and hard about your limitations as a guide to ranges for your shots.
Good luck.
FH