Help, my mum wants to let our cats outside!

We live in a neighbourhood which has this long row of back gardens, making it separate from the road, and my mum wants to let them outside into the garden area. The problem is that I'm sure its not blocked off all the way round the edge, and i have no way of making it so. Alot of people own dogs in my area, and there's a family of foxes on the other side of my road. Plus Archie is a bit dopey and isn't as good at climbing and jumping as his sister, so I'm worried he'll follow her, and get stuck somewhere.
The problem is that both my cats are moggies with a smudge of bengal chucked in so we had no way of knowing what their personalities would be like. I can safely say they are the most active cats me or my parents have ever owned, and we live in a fairly small house. It is possible to keep them inside, but Tilly, despite having several toys to play with, does like to open cupboards, pull everything inside onto the floor then chew it.
In short, we think they need some way to have more space to chase each other and roam around in.
Is there anything that we could use to stop them climbing over the garden fence?
If we could leave the back door open for them it would provide at least a bit more room and having cat grass or maybe catnip to play in would be a good distraction. And it would mean that in summer we'd no longer need metal bars over the windows, or have to shut the cats in the living room whenever we need to hang up the washing.
Any help or ideas whatsoever would be greatly appreciated.

There's a link and ad on this forum for purrfect fence, you could also use a fully enclosed pen (like a large dog pen). If you look up "cat proof fence" in a search engine you may also get suggestions and building ideas. Whatever you choose to do you will still have to supervise your kitties when they're outside. To keep your mother from letting the cats out to run free for the time being you can explain to her all the awful things that could happen to them; dogs, other cats, cars, disease, and cruel people.

As for Tilly's sneaking into the cupboards, look for some baby locks for them.