I've worked on the 408 CheyTac.
While great in concept, in practice it is a royal pain in the rear end.
The brass is horrible. If you like up ten cases, they resemble a police line up. No two shoulders are alike.
Just beware that your initial nights will consist of reworking all the cases, then you better anneal them too.
I like to use molten lead for this.
Now you need a gun and your question answered.
No way is a Remington going to work for this cartridge. The case is just too big for the bolt. No safe way to do it anyways, not in my opinion.
I'd suggest a Nesika Model M. They won't do it for you, so you'll have to take to to someone who will. You need the M cause its long enough (with a little port elongating mind you ) and has the bigger bolt diameter which is what you REALLY need here. This case is a boomer for sure.
If you really get serious about it, please gimme a shout. I'd be happy to help if I could
Awesome round. Good luck.