There is actually more than one reason why the code above fails. Let's break it down a little and explore a few options on how to fix it.
First let's remove the new and try building an instance of A directly in main, so that you see that the first part of the problem has nothing to do with lifetimes:
struct C;
struct B<'b> {
    c: &'b C,
}
struct A<'a> {
    b: B<'a>,
    c: C,
}
fn main() {
    // I copied your new directly here
    // and renamed c1 so we know what "c"
    // the errors refer to
    let c1 = C;
    let _ = A {
        c: c1,
        b: B { c: &c1 },
    };
}
this fails with:
error[E0382]: use of moved value: `c1`
  --> src/main.rs:20:20
   |
19 |         c: c1,
   |            -- value moved here
20 |         b: B { c: &c1 },
   |                    ^^ value used here after move
   |
   = note: move occurs because `c1` has type `C`, which does not implement the `Copy` trait
what it says is that if you assign c1 to c, you move its ownership to c (i.e. you can't access it any longer through c1, only through c). This means that all the references to c1 would be no longer valid. But you have a &c1 still in scope (in B), so the compiler can't let you compile this code.
The compiler hints at a possible solution in the error message when it says that type C is non-copyable. If you could make a copy of a C, your code would then be valid, because assigning c1 to c would create a new copy of the value instead of moving ownership of the original copy.
We can make C copyable by changing its definition like this:
#[derive(Copy, Clone)]
struct C;
Now the code above works. Note that what @matthieu-m comments is still true: we can't store both the reference to a value and the value itself in B (we're storing a reference to a value and a COPY of the value here). That's not just for structs, though, it's how ownership works.
Now, if you don't want to (or can't) make C copyable, you can store references in both A and B instead.
struct C;
struct B<'b> {
    c: &'b C,
}
struct A<'a> {
    b: B<'a>,
    c: &'a C, // now this is a reference too
}
fn main() {
    let c1 = C;
    let _ = A {
        c: &c1,
        b: B { c: &c1 },
    };
}
All good then? Not really... we still want to move the creation of A back into a new method. And THAT's where we will run in trouble with lifetimes. Let's move the creation of A back into a method:
impl<'a> A<'a> {
    fn new() -> A<'a> {
        let c1 = C;
        A {
            c: &c1,
            b: B { c: &c1 },
        }
    }
}
as anticipated, here's our lifetime error:
error[E0597]: `c1` does not live long enough
  --> src/main.rs:17:17
   |
17 |             c: &c1,
   |                 ^^ borrowed value does not live long enough
...
20 |     }
   |     - borrowed value only lives until here
   |
note: borrowed value must be valid for the lifetime 'a as defined on the impl at 13:1...
  --> src/main.rs:13:1
   |
13 | impl<'a> A<'a> {
   | ^^^^^^^^^^^^^^
error[E0597]: `c1` does not live long enough
  --> src/main.rs:18:24
   |
18 |             b: B { c: &c1 },
   |                        ^^ borrowed value does not live long enough
19 |         }
20 |     }
   |     - borrowed value only lives until here
   |
note: borrowed value must be valid for the lifetime 'a as defined on the impl at 13:1...
  --> src/main.rs:13:1
   |
13 | impl<'a> A<'a> {
   | ^^^^^^^^^^^^^^
this is because c1 is destroyed at the end of the new method, so we can't return a reference to it.
fn new() -> A<'a> {
    let c1 = C; // we create c1 here
    A {
        c: &c1,          // ...take a reference to it
        b: B { c: &c1 }, // ...and another
    }
} // and destroy c1 here (so we can't return A with a reference to c1)
One possible solution is to create C outside of new and pass it in as a parameter:
struct C;
struct B<'b> {
    c: &'b C,
}
struct A<'a> {
    b: B<'a>,
    c: &'a C
}
fn main() {
    let c1 = C;
    let _ = A::new(&c1);
}
impl<'a> A<'a> {
    fn new(c: &'a C) -> A<'a> {
        A {c: c, b: B{c: c}}
    }
}
playground