You can, BUT the remington 700 action and bolt are marginal for the 338. L.M. By the time you but the rifle, throw everything else away, then true up the action. You are better off Just buying an action made for it. I have a Stiller Tac338 action at smith right now being built. I put a Kreiger barrel on it. Problem with the Stiller action, is it is longer and a little beefier than the Remmy. So stock options are small. I went with a McMillian A-5, and the smith, has to modify it to fit. Surgeon makes a really nice action for 338. About 300.00 to 500.00 more bucks though.
If I was to start over. I would buy a Remmy 700 in 338 Rum. It is very close to the Laupua round. Factory right off the shelf gun. Biggest down side is the brass. The 338 Rum only option is Remmington brass compared to Lapua brass on the 338 L.M. Or you can buy a Remmington 700 rifle in 300 Rum rebarrell to 338 Edge. Uses the cheaper, both in price and quality of the 338 Rum.
338 Lapua is not a cheap round to shoot. 2.50 to 3.00 per round for brass. .50 to 1.00 per round for the bullet and figure only about 80 to 100 rounds per pound of powder.