The Kayak is just a means to move you from one location to another. Just as hiking (backpacking), car camping, and mountain bike camping all use something to move you and carry your stuff.
The interior available space within many kayaks is greater that the average 60 liter backpack that people use for multiple days in the woods. The type of kayak you should look at would be one that has two bulkheads one in the bow and one in the stern. Having a closed storage in the front and read of the kayak will also add buoyancy to the kayak in the event of it capsizing.
Another good option is the sit on top kayak. There is so much space in that boat it is crazy. You just need to use small dry bags that have labeled long ropes attached in case they move around. This way you will not strain your back trying to look for gear.
It is all about making choices of balancing your skill, taste, budget, and comfort at night.