You could do that but you would need to bring them out into the light in spring. The seeds may also benefit from some snow cover which they won't have in the shed. And with that little soil you won't have very healthy seedlings. You need a container with a depth of 3-4 inches of soil for good roots. With so little soil you'll also need to water maybe every day. All in all, I'll say not a great idea.

Berger,
Did you read the FAQ? They do a great job of explaining the process and why it's important to have a certain depth of soil and good drainage as well as top vents to allow snow and rain to moisten the soil. We leave the containers outdoors so the seeds can benefit from all that Mother Nature has provided, but the containers give some extra protection and keep seeds organized and (hopefully) labeled.

Martha