He tries to sleep, really, he does, but the ideas race through his head nonstop, making it impossible for slumber to claim him. He begins to map out blueprints in his head before he remembers he should be sleeping because it's long past midnight and he hasn't slept for at least thirty two hours. Tossing and turning, he tries to stem the flow of creativity just long enough to get a precious few hours of rest. He throws off the suddenly stifling blankets, but then his arc reactor casts a blue glow on the room that only makes it easier to stay awake. He yawns and shifts and itches and scratches and fidgets and stares at the ceiling and he tries to fall asleep.
He tries to sleep, but can't. So he sighs, annoyed at his uncooperative idea-filled brain, and gets up, and goes back down to the lab where he works on his projects. He invents, and makes amazing, almost surreal machines, but he cannot sleep.
He wishes he could have shot Loki, sent an arrow straight through his heart. (if he even has one) He knows that letting the demi-god go back to Asgard was what he had to do, his only course of action, but that doesn't change the fact that all he can see when he closes his eyes are fragmented images from when he was under Loki's control and he still can feel that sensation of helplessness that comes from knowing you are hurting people you care about and there's nothing you can do to stop yourself. He remembers screaming inside his mind and still being unable to make his own body obey his commands, and wishes he would have shot the man responsible.
He wishes he had shot Loki, but he didn't. So he aims, draws back his bow, and fires, and he imagines each target to be the one he truly wants to hit. He shoots, and his arrows always hit their target, but he cannot sleep.
He wants to go back to his own time, to when things made sense, and see his friends again, have that dance with Peggy. He knows it won't happen though, and he's almost made his peace with that. He may not have the Howling Commandos anymore but he has the Avengers, this strange group of people who have for some reason chosen him, the metaphorical fish out of proverbial water, to lead them. He has them, and that's why he can't sleep. He's irrationally afraid that if he does, when he wakes up, they will be gone and he will be alone again and more than anything he just wants to be back in his rightful time, when he didn't have to worry about all this.
He wants to go back in time, but that's impossible. So he stays in the gym and hits the punching bag, grateful for his new friends and missing his old ones. He punches, and demolishes punching bags a normal man could barely move, but he cannot sleep.
He worries that someday one of his team will get themselves killed, because he's all too well aware how fragile these Midguardians and their things are. Sometimes he just doesn't understand how these humans can be so brave when they could be crushed at any moment, he never doubts for a moment that they are brave, though. Of course they are. They throw themselves at the danger fighting to make the world a better, safer, place and really he's very proud of them. They come back with bruises, cuts and scrapes, and sometimes one of them will get seriously injured and the rest of the team will wait with baited breath for them to recover and they always do but he worries that one day someone won't and they will be left mourning the loss of a comrade, a friend.
He worries that his friends will die, and they could. So he walks back and forth on the roof, his favorite place to come, twirling moljnir in his fingers and worrying. He paces, and could at any moment call lightning out of the cloudless sky, but he cannot sleep.
She pretends that she's fine, that everything is fine and dandy, and she's not broken. Not shattered into a million un-reparable pieces. It's not just the death of Phil, a man who was like a father to her. It's everything else and suddenly her nightmares are back, and worse than ever. She tries not to sleep but sometimes she's just too tired and she can't help it, but then she wakes up an hour later covered in sweat, heart pumping, adrenaling racing through her veins, and has to remind herself it was all dream but sometimes it's hard because it felt so real and only half of it is her imagination. The other half is memory. And each morning she gets up and hides the bags under her eyes and smiles and pretends to everyone, including herself, that she's perfectly okay.
She pretends she's fine, but she's not. So she sits in her bed clutching her gun and counting all the weapons she has and how fast she could get to each of them, waiting for the sun to rise. She waits, and knows that if anything happened she could have twelve different weapons within seven seconds, but she cannot sleep.
They should be wary of the Other Guy. Bruce doesn't outright say this usually, but it's there in the way he never sits right next to anyone, doesn't look them in the eye, and doesn't let anyone touch him. He knows that his alter ego could show up and hurt or even kill them at any given moment. He doesn't want anything to happen to his newfound friends because of him so he doesn't let his guard down. He stays awake and watchful because he's afraid if he doesn't something will set him off. He can't afford to sleep because they poke him and punch him and hug him and sit right next to him and generally ignore the fact that he could kill them all if he gets angry enough. He says so sometimes and they just ignore him, but they should always be watchful around him.
They should be wary, but they aren't. So he is for them, never relaxing or letting his guard down and always looking out for anything that could set him off. He watches, and he could destroy a whole city block in a fit of temper, but he cannot sleep.
They can save the world, and have, numerous times, but they cannot sleep.