For home-type construction, this is a new type of concrete 'application' for me. I'm sure you can support a concrete slab to build a basement underneath. You didn't mention how big a garage you want and how big a basement you want but for me, even going with 12' x 20', that's daring. A timber foundation with a slab to support is daring in my book. What kind of 'load' is going in there besides the roof and the walls?
Anyway, those are just my thoughts. If you've seen barns with this type of installation, I wonder what it looks like. My best advice is to consult a structural engineer because these days, there are codes to follow. Who knows what was 'missed' when the barn was built. It does sound interesting and, again, I'm sure it can be done but with the proper, professional advice. I wish I could make your day. My best to ya and hope this helps.