Which brand of cordless phone will permit multiple base units, and will they work together? We have a farm and an office attached to the house - but stone walls between. We've never been able to get reception on a cordless in the office or barn with a base unit in the house.

 Uniden makes models that support registering one handset to multiple bases (as well as multiple handsets to one base, of course.) http://www.uniden.com/products/productdetail.cfm?product=DECT1060 http://www.uniden.com/products/productdetail.cfm?product=DECT1080-2    

Cordless phones are a great invention, but as you're finding out, they do have their limitations.
The best thing to do with your base units are to put them closer to the wall you're trying to transmit through. In other words, put your base unit right against the wall that separates your office and house, and maybe the signal will come through.
If the walls are too thick and it won't, however, there is nothing wrong with getting other cordless phone units to install in your office and down in your barn, if you have the phone capability in there as well. You probably won't be able to take your house phone and charge it in your office phone's charger, because these units work on different frequencies, but you will also be able to answer your phone any time it rings, in any part of your house. We have a three bedroom house, and through the years, we've ended up with different cordless phones in all of them..and they all work fine with no interference.
Personally, I like the Uniden cordless/ corded phone shown below. It's always good to have a corded phone somewhere in your house in case of a power shortage, and this combines a cordless and a corded phone.