If you are looking for solid sub moa performance with handloads, MANY factory rifles will do that with a bit of tuning ie proper bedding and handloading.
Quite a number of Rems and Savage and Tikkas will shoot as low as 1/2 MOA out to 300yds and sub MOA out beyond 1000yds on a consistent basis. Is that accurate enough?
I have shot factory Savage HB's in 223 out to a mile on MOA sized boulders and hit.
The reason most of us 'build' a rifle is to either create something we can't buy or to gain a performance advantage over what is on the shelf. Budgets can vary over an enourmous range yet group sizes aren't much different.
So take a look at the many offerings from all the major rifle manfs. I am a huge Savage fan and feel these offer the most options for the best value. They certainly need a bit of TLC but return performance well beyond their price point. Are easily modified and when done, shoot with the rigs costing several times more.
One of the most consistently accurate factory rifles I have shot and seen shoot at LR are the Tikka Varminters. I have seen them shoot groups that mega dollar bench rest rigs would be happy to achieve at 1000m.
There is little reason why you can't have a great shooting rifle AND optics for the budget you have in mind.
Here is something I put together on the cheap but shoots really well at LR.
http://www.longrangehunting.com/articles/savage-tactical-rifle-project-1.php
Because I built the stock myself, it is on par cost wise with some factory rifles.
http://www.longrangehunting.com/articles/long-range-rig.php
Should answer your questions on how to put together a rig without breaking the bank.
Enjoy...
Jerry