Don't see why you can't make oaked meat. Corned beef and salt pork have been around for a long time. The 'corn' referred to isn't actually corn, but is rock salt. Here is a good starting point if you want to make oaked, corned beef which doesn't require refrigeration:
In the above article it specifically mentions barrels for storing the salted meat.
Try google with [salting meat].
Having read some research some years ago regarding the propagation of bacteria on artificial (plastic) and natural (wood) cutting boards, and the large difference in the ability of bacteria to grow on these two mediums, wood being by far the most bacteria resistant, it would seem at least prudent and in your case, desirable, to use wood containers. Oak barrels are available in many sizes and you should be able to order them with removable covers with a bit of inquiry.
There is plenty of oak, in many forms, commercially available. The explosion of wineries has produced many caterers to the makers of that product.
Here is a source for oak chips:
Here is another source for all types of oak, including barrels:
If you don't want to make the commitment of barrels, months long curing and the space requirements for same, you could simply buy some winery chips, brine your meat in the refrigerator with oak chips between layers of meat or surrounding a piece of meat.