I'm surprised no one has mentioned a 1909 Argentine (marked "Berlin") Mauser. Many wonderful custom guns have been built up on this action, and many have been rebarreled to 30-06. You can get into one for ~$400 in good condition with basic wood... use the budget balance to upgrade the stock.
Here is an example of one.
https://www.gunbroker.com/item/838073328 If the bore is good (ask the seller), probably the first two things I would do is shorten and recrown the barrel and install some medium height base-and-rings with a decent scope. If it shoots to your liking, put the next dollar into a blue job and start shopping for a stock.
