Here's what I did -
Buy a used Remington 700 SPS Tactical .308 that had maybe 20 rounds through it - it was pristine. It's already extremely accurate as is, but it went from .80-1 MOA to .30-.50 MOA just with a couple mods. I like the 20" barrel on it, but I've been considering cutting it down to 16" and putting a brake on it, although I'd rather get some good glass before doing that.
I bought a Bell and Carlson A2 stock and a Timney 510. I haven't bedded, trued, lapped or blueprinted it yet, and am using the stock barrel. On a good day, I can put 5 rounds through almost the same hole, and on a bad day I can put five rounds inside a dime at 100yds. I also added a Atlasworxs DBM kit last November that I picked up in a group buy, and it makes things a lot easier, but is certainly not needed.
All in all, I have about $1,300 into the rifle without a scope. I had a Leupold Mark 4 on it, but sold it due to needing funds. It currently has a $60 Wal-Mart Center-Point on it, but it's not a bad scope and allows me to shoot out to 500yds no problem. In the next couple months I'm going to pick up one of the $200 Primary Arms 4-16x44 scopes to throw on it, as they're great scopes for the money. Here's a photo of it right after I put the DBM kit on it a year ago:
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And this is 6 shots at 500yds with 3-5MPH winds.
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If money is of no concern: Surgeon Action, Hart or Krieger barrel chambered in 338 Lapua\300 Win Mag\6.5 Creedmore, CADEX Defense stock, Timney trigger, US Optics glass. :laugh:
You never really mentioned what kind of hunting or competition, though. Bears require a little more juice than a deer, which requires a little more oomph than a squirrel - on top of that, most of your competition rifles weigh a ton and aren't that great for hunting, especially considering if you plan on doing 500yds+, you're not going to have a 3-9x scope on it. Likewise, F-Class would require a different rifle than NRA High Power, which is a different set-up than benchrest shooting.
We could sit here and talk calibers all day, but it really depends on a plethora of variables, including but certainly not limited to: Do you reload? If not you should be. How much money do you plan on putting into this? What other calibers do you currently own?
A .308 will get you out to 1,000yds, hell I've sent a .223 out that far pretty accurately. Will a .338 Lapua get out there easier? How about a .416 Barrett, or a .50BMG, or a .300 Win mag, or a 6.5 Creedmore, or a 6.5-06 A-Square, or a 6.5-284 Norma, or a .243 AI, or a 6mmXC, or a .260 Remington (6.5-08 A-Square).. the list can go on and on of calibers that are good for 1000yds - but did you notice a common denominator? Mainly, they're based of 6.5mm bullets. Check out
http://www.6mmbr.com/index.html for more into on that. A .308 will also do extremely well within 500yds as well, but again there's more accurate cartridges.
You're going to find no matter what you decide competition wise, it's probably going to be in the 6 mm, 6.5 mm, 7 mm, or .30 caliber chamberings. Now here's my question about reloading coming into play; you're not going to walk into ****'s or Wal-Mart and pick up a box of 6.5x55 Swede, and if you try to buy them online you are going to be paying through the roof.
With all of this said, it's hard to beat a Remington 700. In my personal opinion, the best models that are going to accomplish both hunting and competition well are the SPS Tactical, the 700P, and the 5R. The SPS Tactical comes in at half the price of the other two (only reason I bought it over a 700P\5R was I found a smoking deal on it used), but the stock is garbage and will need to be immediately replaced if you want to do anything serious.