How to do corners of shed siding using fence pickets

T-111 is pretty much the standard for that kind of stuff. What kind of a look are you going for? You could always sheath it with OSB and put anything you want on top of that (vinyl, clapboard, etc.)

If you look closely at the boards on the inside, you will see they are overlapping, the top edge being behind the bottom edge of the board above (behind when viewed from the outside). This is standard clapboard type construction (although many modern clapboards have a tapered upper edge and the overlap is much greater). Rain tends to drip off the lower edge of the board and proceed downward, not into the joints.

The doors are probably tongue in groove which would resist moisture penetration by creating a circuitous path. Also, moisture that gets into the joint tends to cause it to swell and further seal the joint (somewhat).

The roof joints largely depend on overhang to keep water out.

In general, falling rain not whipped by high winds would largely stay outside.

However this type of construction is not meant to be very watertight, only keep the large majority of the water out. If you have sensitive materials that must be kept very dry, you would probably need to line the shed with a real membrane seal.

If you just want a little increased protection, you could seal the joints with caulk or silicone sealant. Most sealants prefer painted surfaces, so if you are not finishing the surface, look for one that says it bonds to wood.