One thing I can tell you from experience:
The .308 case based cartridges certainly can be made to feed properly through a 98 action,but do not count on just chambering the rifle and having it work.
It will likely need work to get it to feed.
If you are a handloader,The 8x57 is a great choice.Not so many bullets,but,how many do you need?
With the lighter bullets(210 grain or so) I think a .338-06 would work out well in a 98 standard action.Quite standard for dies and reamers.
IMO,a 8x57 case necked up to .338 would be a great wildcat(no,not a hotrod,just a great hunting round) but it not on anyones list,custom dies and reamers make it impractical.
Another kinky possibility,get a .30 cal bbl.You short chamber the bbl with a 30-06 die to 8x57 length.Then shorten the bottom off a set of 30-06 dies to properly size and load the cartridges to that chamber.
An old smith I know has made several of these.He calls them 30 Dunlap.
Its an old school way,gets a match tight chamber from taking 5mm length of the case taper.