Mushy Rigidbody

I want to set up a dome that is semi-rigid, so it holds its structure, but you can still walk through it if you get a running start. Are there any methods of this?

Also so the structure bends accordingly.

That's a pretty complex physics simulation you're trying to do, there's no simple answer to this. You can look at procedural mesh deformation to get you started.

http://unity3d.com/support/resources/example-projects/procedural-examples