The problem with the Lee Enfield, it can be very difficult to get accurate, it depends on the individual example. I've had Lee Enfield's in my workshop that were impossible to get a group smaller then 24 inches at 100 yards, while others are very tight groupers.
What has destroyed so many Lee Enfield's is issue cordite ammunition with corrosive primers, cordite burns far hotter then other propellent's leading to erosion of the chamber throat. IF you decide on a Lee Enfield, check out the barrel very carefully.
A better rifle in my opinion is the Enfield P14 or P17 built by Winchester in 303 British and 30-06 respectively, quite a few have been rebarrelled in South Africa to take the 375 H&H, its a very strong and robust action.
As far as I am aware, all P14 rifles converted to sniper rifles were Winchester built.