How can I fill the gaps between barn wood siding boards?
Do not try to use caulk between all the boards. It will look like crap and on top of that, like you already indicated, it will be hard to put in there in the first place. 

The proper approach to this is to install vertical strips of wood over the cracks from the outside. You can choose to use something like 1x2 inch cedar strips. Or if your shed looks like it is 100 years old and you want to maintain that look you can search out a source of recycled barn boards and rip them into strips to use on your shed. 

The vertical strips are called batten strips. Often folks use even wider strips that may be 1x3". Here is how a typical structure might look: