To purcahse ammunition you will need a Permis de Chasse. Either a current one or the one from the previous year.
French citizens need to pass an exam to get this. Foreigners don't but evidence of their "competence". An FAC giving use of the rifle for deer or other live quarry in the UK should suffice for that.
The Permis de Chasse can be for a year, for nine days, for three days. The last two can only be issued two times in the same year which runs from 1st July.
You pay more for the longer period, of course, AND you have to decide if you want a Permis de Chasse just for that "department" (like our counties) or one that is nationwide.
Firearms are allowed on the Eurostar but ammunition is prohibited. I don't know about Eurotunnel.
Lastly, like so much of France, things "at night" are illegal! Shooting (and fishing) are illegal after dusk so no "boar under the moon"...except in the three departments that border Germany where they have German hunting laws because it suits them! French on 14th July yet still German when they want something else!
Some places restrict certain public forests to those resident in the "commune" (a community or group of villages or district).
You can get a Permis de Chasse by post or in person or I think now by internet. They will want the evidence of "competence" AND proof that you have insurance issued by an approved French insurance company for shooting cover.
BASC or SG or whatever other cover will not suffice AFAIK.
Here is the website of the relevant Government agency:
http://www.oncfs.gouv.fr
As others say weapons in military calibres are prohibited. That means actually in French law any weapon "capable of firing ammunition that can be used in a military weapon" so .275 Rigby is banned!
But oddly no problem with 8x57JRS or 7x57JR (the rimmed versions for double rifles or combination guns). Double rifles have a limited following, very rare to see any sort of drilling or bockbuscheflinte, and self-loading rifles are widely used.
The effect of that law is that even if you did, illegally, take a 308 to France you would not be able to buy, off the shelf, sporting ammunition for it as gun shops won't stock what there isn't (because of that military calibre ban) a market for.
Your 7mm-08 is no problem and is widely popular in France in fact...as is 270 Winchester and (in self-loading rifles) 280 Remington and 300 Winchester Magnum. So it's not just a land of 7x64 although that has a certain popularity.
In fact in most good gun shops you'll find a far wider selection of calibres that in most gun shops in UK.