Seems to me that you'll need to narrow down what you expect from the "precision rifle". If it's just pure accuracy, and 3/4 to 1/2 MOA is your goal, then you're going down the right path (if there is more or different requirements, you may be better off on another path). Be prepared to do A LOT of work (and have the tools and ability) to make it happen...

Look for a bargain in a Stevens 200 or Savage Model 10 action. Replace the trigger with something that will give you a consistent release (I think you can do well with the Rifle Basix and Fred from SSS has the competition trigger which can get you down to 12 oz). Find a good deal on a nice barrel (if you can swing it, I'd go Criterion - but you CAN find decent factory savage target barrels here). It is feasible to "build" an accurate stock from a Boyds (~ $110 + time and materials to pillar bed the action to the stock). Make sure you get the whole thing PERFECTLY square and concentric from back to front - ba-da-bing ba-da-bang... you'll have a real nice accurate semi-custom rifle. PS - don't forget that you'll need to build your own loads for this rifle if you want to get it to be closer to that 1/2 MOA target.

Keep in mind there are more options on the Savage path that make all this easier (less work), but then again, you'll need to adapt your "low dollar" criteria by a good bit. For example, one of my fav. stocks that'll give you a great platform for accuracy is the HS Precision stock. Hell... just go find a nice Model 12 target rifle and you've got what you need straight from the factory. One thing you MUST be willing to accept... all Savages have a heavy bolt-lift (some better than others, and you can work on that too to some extent - Fred at SSS can REALLY help with that).

Good luck and best wishes on your project - it's loads of fun getting there!