It sounds to me like you want to rely on quantifiable science in all things. Yes science is founded upon math which is founded upon logic. Both math and logic are axiomatic systems, and axioms are assumed or declared. They are not independent of a person's mind either. You cannot escape it. David Hume sorted it out for us long ago. Rational certainty is a pipe dream. But that need not prevent us from navigating this world and enjoying life.
Back to the topic: we seem to be talking about a human being choosing between competing news sources. As that consumer, you get to define "unbiased" for yourself. Wise people have given it a lot of thought, so you might consider consulting what they said.
And you can draw the line at something being measurable to matter to you. That's your choice. I have found the things that matter most are fictions in my brain: love comes to mind.
It is also not necessary to measure to detect a distance. I love my dog, and I love my child. I cannot measure love, yet I detect a significant distance between my love for my dog and my love for my child.
Like love, "bias" has meaning even if it cannot be measured. Like love, one can detect distances between biases, even if they cannot be measured. Science (measurement), math (quantity), and logic (relations of ideas) are just as ephemeral, because they are founded upon unprovable axioms and norms.