The Talmud in Gitin 6b by the story of pelegesh begeva says 2 opinions what the problem was. Rav Evyosor says it was a fly, and Rav Yonason says it was a hair. One day Rav Evyosor met Eliyahu and asked him "what is G-d doing now?" So he answered "he's learning the sugya of pelegesh begeva." "And what is he saying?" He's saying "Evyosor my son says like this, and Yonason my son says like this." So he asked Eliyahu "can there be uncertainty in heaven?" (Doesn't G-d know the facts?) So he answered "eylu ve eylu devray elokim chayim." They're both true. First there was a fly and then a hair."
You see clearly from here that it's only eylu ve eylu because they both happened, but if we knew for a fact that only 1 thing happened, then one of them would have been making a mistake, and G-d would not have mentioned him.
In Ksubos 57a we have 2 options how to learn up a complicated machlokis. Option 1 is that Rav Yochanon and Rebbe Yehoshuah ben levi argue about what Rebbe Yehuda and Rebbe Yosi say. And option 2 is that Rav Deme and Rovin argue about what Rav Yochanon and Rebbe Yehoshuah ben levi say. So Rav popa says that he wishes he can have option 1 rather then option 2, because option 1 is a 'svara' argument, while option 2 is a 'shmua' argument.
In other words, Rav Yochanon and Rebbe Yehoshua ben levi never talked to Rebbe Yehuda and Rebbe Yosi.They only came to their conclusions through 'svara', and you can always say "eylu ve eylu divray elokim chayim". And if someones 'svara' does not apply well in this case it can always apply well in a slightly different case. But by 'shmua' (He heard it from his rebbe) what did the rebbe say? Did he say this or that? Somebody has to be wrong, and you don't say eylu ve eylu.(See Rashi, Tosfos, and the hakdama to Yam shell Shloma, Baba Kama)
Rav Yochanon was sitting and teaching that G-d will (In the future) bring precious stones and pearls as big as 50 x 50 ft. and put them by the gates of Jerusalem. So one of his students made fun of him. "You can't even find a precious stone as big as a little birds egg, and now you'll find 50 x 50? One day the student was at sea and he saw the angels cutting these huge stones, so he came back to Rav Yochanon and said "you're doing good, keep on going, I saw it with my own eyes". So Rav Yochanon said "You good for nothing, if you would not have seen it you would never have believed me. You're making fun of the words of the rabbis!"
What did Rav Yochanon do? He gave him the "eyes" and he turned him into a heap of bones (He killed him) (Sanhedrin 100a) If you make fun of the words of the rabbis you get killed (Eruvin 21b)
Now what kind of science did this student engage Rav Yochanon with? It was simple arrogant science, "I never heard of such a thing, therefor it cannot be true". A lot of science pitted against chazal is this kind of garbage. "I never heard of such a creature, and National Geographic never talked about it, therefor it doesn't exist." Let me ask you a question. How come National Geographic never talks about Noah's ark on top of Mt. Ararat (which they know very well about) and how come the CIA, the KGB, and the Mossad never release the hundreds of fly-by pictures they took of Noah's ark? Is it because they don't want you to believe in the Torah? hmmmmmm. I never thought of that!
Then there are some scientists that are so arrogant that they make huge fools out of themselves. Click Nov 3 and you'll see what I mean. And of course all the scientists who don't believe in G-d are outright liars, because they see all the super intelligence of the world all around them, and they say that it came by itself. So they have a chazaka of being a liar.
I could go on and on but let me just quote rabbi Urin Reich. "They don't even know what the weather will be like in two weeks but they know what happened 65 million years ago".
That said, we all have to agree that there is still a lot of good 'rock solid' science out there. Like let's say Rav Yochanon would have darshind from a posuk that the world is flat (Let's just say) and his student would take him for a plane ride at 100,000 feet, there is no question that Rav Yochanon would have said "I made a mistake". (Showing pictures of the earth is not good enough. It can always be a fake) We also find in Sukka 8 that the Rabonon of Kaysrey made a mistake in geometry.
In Yoma10 there are 2 versions what will happen at the end of days (Today). One is that Rome will fall to Persia (Iran), and one is that Persia will fall to Rome. There is no checkmate pushing you to say that only one of them is true and there is no eylu ve eylu. It can very well be that they will both happen, and indeed this is what the Malbim says, like I showed you in Jan 30. (Click)
P.S. unless you want to say that if the earth is really round Rav Yochanon would never Darshen that it's flat. It just wouldn't happen. The same thing with Gitin 6b. If they both weren't true they both would not have darshend, Only one. I don't know.
P.S. You also have in the end of Chullin; 141a/b that many Tanaim may have misunderstood certain reasoning's, and because of this they created false Brysos (Like "Rabbe Yehuda said this and this", but that's not really what he said). And that's why any Brysa that was not said by Rabbe Cheya and Rabbe Osheya is not reliable. And it goes without saying that it would not be part of eylu ve eylu. But this problem of misunderstanding is only in Halacha, but Medrash would be a different story.