Demanding impossible evidence?

I seems like they know they cannot discredit or do not want to accept the source so they try to discredit the person giving the source.

 red herring, ad hominem, Cognitive dissonance

Two great points here: 1) evidence demanded is not impossible and 2) what is demanded wouldn't reasonably be considered "evidence" for the actual claim.

Interesting, I didn't think it was a red herring because the evidence they were demanding was related to the claim (even if I, too, consider swimming in Fukushima's reactor pools to be weak evidence of safety). 

I like Dr. Bennett's distinction above about an unreasonable demand, as opposed to impossible , since as you mentioned it isn't impossible, but certainly given that it's prohibited, it does seem unreasonable. (In particular because it's not that anyone swims in the water, but rather that I do.)

The claim is there is significant scientific evidence that Fukushima's waters are "generally" safe. One presumes that means that the radiation levels are sufficiently low as not to present any immediate harm. 

Likewise, every year a few visitors to Georgia State's pristine lakes and river systems are infected with Necrotizing fasciitis. Swimming in them is generally safe, but not altogether without significant danger. 

The respondent argues evidence be damned, he would only be convinced if his opponent actually bathed in Fukushima's waters. That's the deception.

His argument implies that determining lethal or harmless radiation levels and "general safety" of Fukushima's waters can be measured by his opponent physically swimming in it. Which is utter nonsense. This sleight of hand redirect leads us nowhere closer to the truth, and thus remains a big, stinking, flopping red herring. 

I elided the vast majority of the conversation to focus on what I think is relevant to my question. However, it's not removing much: suffice it to say, they did not offer any justification for rejecting my sources (or, for that matter, any other part of the argument I was making using those sources). When I asked specifically about this, they stated they wouldn't read them until I "put my money where my mouth is" by swimming at Fukushima. 

Maybe they had good reasons for rejecting my sources or argument, but they didn't offer them to me, so I could only speculate. 

Sometimes the evidence exists, but cannot be dug up on a short notice, and should not be required.

Lets say you are sitting in a bus stop, someone argues with you about the Earth. You say the earth is round and revolves around the sun. The other guys might say you have to prove it. How do you provide evidence to well known facts when sitting in a bus stop?

Other example is you just got a new job at a company, sitting on a project meeting, someone proposes to do an engineering process on a new device, that they used to do on old obsolete devices. Then you say we cannot do that because XYZ will happen. They dont believe you, they demand proof of this XYZ happening, they never heard of it. Since you worked at other companies before where XYZ prevention strategies were solidified and applied daily by multiple engineers, you read books about it, you attended conferences on it 10 years ago where multiples of your competitors discussed their similar XYZ prevention strategies, it is a routine. Now at the new company they want you to dig up the conference papers (that you may only have on your personal laptop at home, cannot get it and reread it on this meeting) and the old trial runs you did with your former colleagues 10 years ago (that you don't have access to) to prove it, otherwise the team is full steam ahead and ignoring alleged XYZ. The meeting concludes to go full steam ahead, and note in the minutes that you interfered with the experts by talking about made up stuff. This actually happened to me multiple times.

What is causing it, is the listener to your claim being unexpectedly uneducated. Or dealing with fake experts. A fake expert is someone with less than average knowledge, who was promoted to being lead engineers simply through length of service, but claim to "know everything there is worth knowing in the field". The opposite of an open mind.

I have to admit I do not understand your point.