Check out Larry's
Tiny Trailer Workshop. It's amazing how much you can fit into a small space.
My shed is 20'x9' and is far too small for my needs.
I'm a compulsive hoarder, and collect odds & ends of wood whenever/wherever possible. If I got rid of all the clutter I could work in a wheelbarrow.
I currently have an Axminster AWVSL1000, which would fit in your shed nicely. You might struggle to get anything else in, mind you.
I resisted going for a smaller lathe because you can turn small stuff on a big lathe, but you can't turn bigger stuff on a small lathe. I would echo the advice though to try a club's lathes before making any decision. I was advised to avoid the Axi lathes because, "they are made in China and the castings are very poor quality." I plumped instead for a Record Power lathe with the twin bar bed system. This lasted less than a year before I became frustrated by the lack of power, vibration in the bars making the headstock wander, the awkwardness of the wheel on the tailstock, and the apparent inability to keep headstock and tailstock in alignment.
Please bear in mind this is a completely personal opinion based on my own experience of different lathe types and is not a definitive guide. I have been told many times my opinion is wrong. No reason why this time should be any different.
Either way, good luck with the hunt, and remember:- No pictures means it didn't happen.