Don't forget to buy some ammo for it; that can be very frustrating. I bought a new pistol last year & had heck finding .45ACP in any quantity.
I don't know if you would get the accuracy you are looking for, but have you considered a military surplus rifle? Some of them look a little old-fashioned, but it could save you some bucks.
Maybe some of the more rifle-savvy members can advise if a milsurp rifle could get you the accuracy you want. The ones i see advertised all seem to have iron sights, which might be more of an optional item on a more modern bolt gun.
look at these for example: Czechoslavakian vz. 24 8mm Mauser Rifles
8mm mauser for a little more than $200. The old Mosin Nagants are even less expensive, but i am under the impression that accuracy can vary greatly from rifle to rifle & from one batch of surplus ammo to another.