First of all, you are better off going cheap on the rifle than the scope. Many modern rifles around $500 or less guarantee sub moa out of the box with factory ammo. You can generally speaking get them down to 1/2 or even 1/4 moa with a well developed hand load.
I recommend the Ruger American Rifle in either.300 Win Mag, or 30-06. The 30-06 is about $100 cheaper but the .300 win mag will launch larger pills at faster velocity with higher ballistic coefficients down range making longer shots easier. And the 300 Win Mag is a .30 cal so bullets are easy to find. I have a Ruger Am rifle in .270 win which I was able to get at 1/4 moa without really trying to develop a good hand load. The Ruger American also comes standard with a free floated cold hammer forged barrel and adjustable 3-5 lb trigger.
The .300 win mag is about $500 and the .30-06 is about $400.
For a scope I would go with the Leupold VX-R 6x18x40 Mildot Or the Nikon Monarch 4x16x50 Mildot mounted using Burris signature XTR rings. The scopes are about $500 each and rings $80. The Burris rings have the ability to have either a 0 MOA cant (like standard rings) or 40 MOA cant. This is necessary when trying to zero scopes with standard amounts of internal travel at 800+ yards. Don't shim your rings....and the XTR rings by Burris will handle heavy recoil and are easy on the scope.
You could put this package together for $1000-$1100 and have a nice 1/2 MOA rifle, decent glass, and crisp light trigger. You will not have iron sights but I have been hunting for decades and was in theater over in the Persian gulf and never once used iron sights. With good rings and a good scope, as long as you don't bend it falling out of a tree your fine. And if you go on a long hunting trip, keep a backup rifle in your truck.
Also, no iron sights allows for medium or sometimes even low rings.