1. What is it?

The impossible choice is when a character is given a choice, usually of two options, in which the consequences of both are awful.

2. How is it made?

3. Examples in literature

Catch-22 
by Joseph Heller

Excerpt from Chapter 5:

It was a horrible joke, but Doc Daneeka didn’t laugh until Yossarian came to him one mission later and pleaded again, without any real expectation of success, to be grounded. Doc Daneeka snickered once and was soon immersed in problems of his own, which included Chief White Halfoat, who had been challenging him all that morning to Indian wrestle, and Yossarian, who decided right then and there to go crazy.
‘You’re wasting your time,’ Doc Daneeka was forced to tell him.
‘Can’t you ground someone who’s crazy?’
‘Oh, sure. I have to. There’s a rule saying I have to ground anyone who’s crazy.’
‘Then why don’t you ground me? I’m crazy. Ask Clevinger.’
‘Clevinger? Where is Clevinger? You find Clevinger and I’ll ask him.’
‘Then ask any of the others. They’ll tell you how crazy I am.’
‘They’re crazy.’
‘Then why don’t you ground them?’
‘Why don’t they ask me to ground them?’
‘Because they’re crazy, that’s why.’
‘Of course they’re crazy,’ Doc Daneeka replied. ‘I just told you they’re crazy, didn’t I? And you can’t let crazy people decide whether you’re crazy or not, can you?’
Yossarian looked at him soberly and tried another approach. ‘Is Orr crazy?’
‘He sure is,’ Doc Daneeka said.
‘Can you ground him?’
‘I sure can. But first he has to ask me to. That’s part of the rule.’
‘Then why doesn’t he ask you to?’
‘Because he’s crazy,’ Doc Daneeka said. ‘He has to be crazy to keep flying combat missions after all the close calls he’s had. Sure, I can ground Orr. But first he has to ask me to.’
‘That’s all he has to do to be grounded?’
‘That’s all. Let him ask me.’
‘And then you can ground him?’ Yossarian asked.
‘No. Then I can’t ground him.’
‘You mean there’s a catch?’
‘Sure there’s a catch,’ Doc Daneeka replied. ‘Catch-22. Anyone who wants to get out of combat duty isn’t really crazy.’
There was only one catch and that was Catch-22, which specified that a concern for one’s own safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn’t, but if he was sane he had to fly them. If he flew them he was crazy and didn’t have to; but if he didn’t want to he was sane and had to. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle.
‘That’s some catch, that Catch-22,’ he observed.
‘It’s the best there is,’ Doc Daneeka agreed.

Sophie’s Choice 
by William Styron

Excerpt from Chapter 13:

The doctor was a little unsteady on his feet. He leaned over for a moment to an enlisted underling with a clipboard and murmured something, meanwhile absorbedly picking his nose. Eva, pressing heavily against Sophie’s legs, began to cry.
“So you believe in Christ the Redeemer?” the doctor said in a thick-tongued but oddly abstract voice, like that of a lecturer examining the delicately shaded facet of a proposition in logic. Then he said something which for an instant was totally mystifying: “Did He not say, ‘Suffer the little children to come unto Me’?” He turned back to her, moving with the twitchy methodicalness of a drunk.
Sophie, with an inanity poised on her tongue and choked with fear, was about to attempt a reply when the doctor said, “You may keep one of your children.”
“Bitte?” said Sophie.
“You may keep one of your children,” he repeated. “The other one will have to go. Which one will you keep?”
“You mean, I have to choose?”
“You’re a Polack, not a Yid. That gives you a privilege—a choice.”
Her thought processes dwindled, ceased. Then she felt her legs crumple. “I can’t choose! I can’t choose!” She began to scream. Oh, how she recalled her own screams! Tormented angels never screeched so loudly above hell’s pandemonium. “Ich kann nicht wahlen!” she screamed.
The doctor was aware of unwanted attention. “Shut up!” he ordered. “Hurry now and choose. Choose, . . . or I’ll send them both over there. Quick!”
She could not believe any of this. She could not believe that she was now kneeling on the hurtful, abrading concrete, drawing her children toward her so smotheringly tight that she felt that their flesh might be engrafted to hers even through layers of clothes. Her disbelief was total, deranged. It was disbelief reflected in the eyes of the gaunt, waxy-skinned young Rottenfuhrer, the doctor’s aide, to whom she inexplicably found herself looking upward in supplication. He appeared stunned, and he returned her gaze with a wide-eyed baffled expression, as if to say: I can’t understand this either.
“Don’t make me choose,” she heard herself plead in a whisper, “I can’t choose.”
“Send them both over there, then,” the doctor said to the aide, “nach links.”
“Mama!” She heard Eva’s thin but soaring cry at the instant that she thrust the child away from her and rose from the concrete with a clumsy stumbling motion. “Take the baby!” she called out. “Take my little girl!”
At this point the aide—with a careful gentleness that Sophie would try without success to forget— tugged at Eva’s hand and led her away into the waiting legion of the damned. She would forever retain a dim impression that the child had continued to look back, beseeching. But because she was now almost completely blinded by salty, thick, copious tears she was spared whatever expression Eva wore, and she was always grateful for that. For in the bleakest honesty of her heart she knew that she would never have been able to tolerate it, driven nearly mad as she was by her last glimpse of that vanishing small form.