How can I make swinging barn doors weather tight?
Without pictures or a description of what you've currently got to make them &quot;airtight&quot;, it's hard to make a specific recommendation, but in general, do the same things you do for French doors on your house:

Weather stripping all around the openings
Weather stripping between the doors
Rubber sweeps at the bottom of the doors

If the doors weren't originally designed for weatherstripping, adding it may make it very difficult to close them, so you may have to either get some very thin rubber or foam gaskets, or, using some of those woodworking tools in your woodworking shop, adapt the doors and/or openings to make room for thicker weatherstripping.
Whatever you do, just make sure that the doors close tightly and that your updates don't prevent the latches from working.