When the Doctor says that he thought Clara dead for a month, he’s not being quirky or avoiding his feelings or even saying how long it felt—he’s aggregating.
In linear time, it probably was the five minutes Clara claims, roughly, but it’s not the length of time between Bonnie’s declaration and the text message that the Doctor is addressing. Or at least not just that.
He’s adding up all the time, every second, minute, hour, and day, that he had ever thought Clara was dead:
All the seconds, scattered throughout the universe, when they’re running for their lives and she gasps what he is terrified was her last breath, not just exhaustion or the cramp in her leg. The specious present of his life since he pieced together who she was after that Victorian Christmas and went off to seclusion to figure her out, thinking her dead twice over. Each minute he banged on her door having found her again only to discover her lying on the floor, possibly dead again, because he was too late. That singular yet infinite instant at his grave when he felt her in every atom and quark and string of his existence, in every Plank and chronon of his time stream, but all he could think of were her innumerable deaths, that she was dying now.
The confused and disassociated interval he woke up from the end of the universe, and didn’t know where she was or if she was alive. Every ticking zeptosecond of the quantum lock as Clara’s brain was scanned for guilt. Those hours spent as the President of Earth where his first order to find her, when he willingly played into Missy plan to know she was still alive, when he thought she might’ve gone to the grave with Danny Pink.
The Christmas night of dream-time intercalated to her lifespan and back again in the simultaneity correction it took him to enter the Time Vortex and remove the dream crab. The agonizing day he threatened the Dalek’s from Davro’s own chair because he’d seen her exterminated. The heart-wrenching moment when he found Clara inside the Dalek Missy urged him to destroy, remembering when last she’d died like this. The boostrap paradox he created to save her, because he wouldn’t lose her again, and the resulting temporal shift of 125 years-worth of worry that he’d gotten it wrong as the stasis chamber waited to revive him. The relative eternity while she was aboard the Mire ship.
Summed into a term she’d comprehend, it’s a month of dread and shock, guilt and loss, fear and regret, that he’s felt over the near millennium of his timeline that he’s know her as her.
And while that might not seem like much in comparison, as a Time Lord, he knows these times acutely and accurately. All of them, each of them, every one. And he can judge them. He could make her a list, she only need to provide a categorizing variable and an order of magnitude.
How many times he felt that way? When and where did they occur? In chronological or progressive order? How long each time lasted? Which of the over 23.6 quadrillion vibrations of the caesium atom hurt the most?
He measures these times, notes them well, lets them drive him mad with worry and concern, because it is better than the alternative. He would rather think, feel, know that she was dead for billions of years than have it be true. He would let grief destroy him if it meant saving her.
His Impossible Girl, his girl who was born to save him, his girl who he will always remember (until he doesn’t). His Clara. He is terrified of her dying, so he will not let her die.