Some time ago Kim, from The Inadvertent Farmer, asked me for some ideas for a rabbit enclosure to use with her raised garden beds. She wanted a way she could let her rabbit have access to her 4x20 raised bed.
Kim's garden bed with her sketch of what she was thinking.
To keep the rabbits in and ground predators out I suggest you enclose the bed with 2 sections of 25 ' long 48 inch high electric fence from Premier 1. I like this fencing a lot. I use it with my chickens and it keeps everything but the hawks out. It is easy to set up, take down, move, etc. If you have a lot of aerial predators you can bridge between the posts across the bed with some fiberglass fence posts and cover it with plastic bird netting. This fencing could also be setup in other places (like on the lawn) and still protect the rabbits. It is a very flexibly option that I use a lot.
For the rabbits housing I would build a light weight frame out of 3/4 inch pvc pipe and cover three sides and the roof with opaque corrugated polycarbonate. You can get all those bits at Lowe's or some other building supply store. I'm not sure I would put a permanent floor in it. Rabbits tend to dig under things, and a floor is an invitation. I'd probably put in a box like a nest box that they could go in to sleep and hide. I would use plastic as much as possible. You will get less chewing and less rot than if you used wood. It will also be quite lite weight. You may need to anchor it with some tent stakes if you have a lot of wind.
This wouldn't be a year round rabbit shelter, but it would be a way to have them out for a good part of the year. I have been told that allowing rabbits to graze outside will make them sick. This hasn't been my experience, but you may want to check with your vet. I know wild rabbits seem to do well enough grazing year round.
Kim, let me know if this was what you were looking for. If not we can try something else. Anyone else who has ideas along these lines, I'd like to hear from you too.