Sniping is a verb not a noun. Any gun can be a sniper rifle so lets call it a precision rifle.
You can get a savage or some sort of rifle already set up but you didnt build it yourself.
CHEAP? Thats relative.
Get a Mauser 98 action, their about 75 bucks in Shotgun news. Open up the bolt face for the mag case and the mag weld to take the 300 cases. Get a heavy coutour barrel, no less then 24 inches, 26 would be better. You can get these already threaded for the mauser action but they'll probably be in 308 or '06. Just run a 300 WM reamer through it and head space it to your action. You can get a target style stock from Boyds pretty reasonable.
Now so far you prices are kept down. Where the cost comes is fitting. You can have the action trued and bolt lapped. Thats gonna start getting pricy if you want it done right. You can come up with a pretty good shooter without those procedures but its better if you can afford it.
Now get something along the lines of the Luppy 10X mildot scope.
I did pretty much all of the above only using a Model 70 winchester action. The total cost of mine was under $200. $80 for the barrel, $65 for the Bishop Stock, and I had bought 6 M-70 Actions for $225. The sights Refield Oly and Weaver T-10, I had. I did all the work myself but I have the machinery and have been building guns for a while. Mine was built for 1000 yard matches so I made it a single shot rifle. Also has a 29 inch barrel. Dont let people tell you you cant use two piece mounts, I use the weaver because I can mount the scope or take it off and put on irons. I've been shooting this gun in 1000 yard matahes for 30 years and it still shoots. Its picture below with the iron sights.