I'll summarize what pretty much everyone will tell you.
You will get mixed opinions on using a Remington action for a 338 Norma, mainly due to the fact that you must use a .585" bolt face (same as a Rigby or .338 Lapua), and some consider that to create too much bolt thrust for a standard Remington 700 action. Others do not consider it to be too much pressure. The fact is, your cutting your room for error down by using a Remington 700 action instead of a custom action, such as a Stiller TAC338 or similar, with a larger (usually .750") bolt diameter. This does not mean that it cannot be done, and does not mean that you cannot have one built, the smith that I use has built many on a Remington action with no issues, and MANY have been built on that action, including a factory Remington 700 chambered in 338 Lapua. So the long and the short of it....you can do it, and have a safely functioning rifle if you load safely, but your safety margin is smaller than if you use a custom .338 dedicated action. How much smaller? I don't know. Someone much smarter than me would have to tell you that. That is a risk you must be willing to take if you build one on a standard Remington action.