The smallest Unix that I've personally used was Minix. It ran fine on a 286 with a 40 MB (not GB) hard drive. I think that system had 2 MB (not GB) of RAM; maybe 4. I probably still have it here somewhere...
you'll probably end up having the same problems with DOS. Take a look at susestudio.com, you can build a pre-configured appliance to meet your exact specifications. With a little work you should also be able to get the wireless networking setup properly as well.