It almost had to be a .280, as there is no way you could seat a .30-'06 in a .270 chamber without a sledge hammer. The .280 wouldn't be easy, either, and I can't believe you didn't use a lot of force to get it in, but if the .270 chamber were a bit sloppy, it might go. (A good lesson in not having ammunition around that you shouldn't use in the gun you are shooting.)
What probably happened is that the larger cartridge was forced in, but there was not enough room for neck expansion, with the result that the pressure went sky high and the result was predictable. (The oversize bullet would have little effect.)
If the base of the cartridge case is still there, a brass rod should be used to drive it out. You say a steel rod broke. Was it a cleaning rod? I can't imagine a steel rod of anywhere near 1/4 inch just breaking. If the base of the case is gone, you will need a broken shell extractor to remove it.
I think from reading your posts, that it would be better to consult a gunsmith. Not only would he be more likely to get the case out without damage to the gun, but he can then determine if there is any damage that can't be repaired or that would compromise the strength of the rifle.
Jim