Fusion rockets might be able to get a sufficiently large spacecraft over 1% c. If you’re instead sending multiple generations of craft and waiting until each succeeds then it’s 1/n %c etc.

Colonization using a massive habitat capable of extreme redundancy and asteroid mining could be a completely viable solution to colonization. But such a structure wouldn’t be light.

Hitting 1% c could very well take megastructures that a civilization would rather spend on redundant craft etc. We can dream, but we’re nowhere close to being able to say what’s actually viable.