You need to look at permitted development rules. The issue you will have is the proportion of the space you have; if its "most of the garden" it would not be allowed if anyone objected.
A friend built exactly what you plan, 9m x 6m and 4m at the ridge. But he was very careful to make it the correct distance from the boundary and measured carefully the volume against available space he was using.
Even so a neighbour complained and planning came to check, but he was correct. Didn't help neighbourly relations mind.
That was a steel building too so I don't think you'll get much leniency there.
Details for PD are readily available online. If you are in a conversation area or area of outstanding natural beauty its different, more strict, again.