Is Elinor unemotional?

[Elinor] had an excellent heart;—her disposition was affectionate, and her feelings were strong [...] Elinor, too, was deeply afflicted; but still she could struggle, she could exert herself.
Readers sometimes envision Elinor as an unemotional, unfeeling character since she seems overshadowed by Marianne's effervescent, emotional personality. However, Austen makes it clear through the narration that Elinor feels as deeply as Marianne.
Elinor, starting back with a look of horror at the sight of [Willoughby], obeyed the first impulse of her heart in turning instantly to quit the room,...
The difference between them is that Elinor has not succumbed, like Marianne has, to the popular cultural conception born of romanticism that extravagance in emotion is laudable. Elinor holds to her belief that what is truly laudable is to govern grief, joy, love and rapture, to not yield to ungoverned emotionalism. Austen explores the question of whether Elinor is right or wrong and presents the answer when Marianne's, somewhat willful, neglect of her health nearly costs her life.

[Marianne said,] "I saw that my own feelings had prepared my sufferings, and that my want of fortitude under them had almost led me to the grave. My illness, I well knew, had been entirely brought on by myself by such negligence of my own health, as I had felt even at the time to be wrong."
Elinor's ability to govern her strong emotions allows her save herself from similar calamity and, perhaps more importantly, allows her to help nurse Marianne back to health when she grows very ill. Elinor's feelings are only governed, not absent or, worse yet, repressed. When Edward reports that he is single and that Robert is married to Lucy, not him,Elinor yields to her emotions with less restraint.
[Elinor] almost ran out of the room, and as soon as the door was closed, burst into tears of joy, which at first she thought would never cease. Edward ... saw her hurry away, and perhaps saw—or even heard, her emotion...
The greatest character difference between Elinor and Marianne, the element that allows for some to envision Elinor as tranquil and unemotional (which is how some actresses portray her in film adaptations), is that while Austen often displays Marianne's emotionality in her words and actions, Elinor's is most often commented upon by the narrator. Here are their different reactions to leaving their childhood home:

[Marianne]: "Dear, dear Norland!" said Marianne, as she wandered alone before the house, on the last evening of their being there; "when shall I cease to regret you!—when learn to feel a home elsewhere!—Oh! happy house, could you know what I suffer in now viewing you from this spot, from whence perhaps I may view you no more!—And you, ye well-known trees!—but you will continue the same.—No leaf will decay because we are removed, nor any branch become motionless although we can observe you no longer!—No; you will continue the same; unconscious of the pleasure or the regret you occasion, and insensible of any change in those who walk under your shade!—But who will remain to enjoy you?"
[Elinor]: What felt Elinor at that moment? Astonishment, that would have been as painful as it was strong, had not an immediate disbelief of the assertion attended it. She turned towards Lucy in silent amazement, unable to divine the reason or object of such a declaration; and though her complexion varied, she stood firm in incredulity, and felt in no danger of an hysterical fit, or a swoon.
In addition, while Marianne is often exasperated with less emotional people and accusatory of Elinor reserve, Elinor attempts to think things through from the other's perspective and to find explanations for their behavior.

[Marianne to Elinor]: "Esteem him! Like him! Cold-hearted Elinor! Oh! worse than cold-hearted! Ashamed of being otherwise. Use those words again, and I will leave the room this moment."
[Elinor about Marianne's remarks]: Elinor started at this declaration, and was sorry for the warmth she had been betrayed into, in speaking of him. She felt that Edward stood very high in her opinion. She believed the regard to be mutual; but she required greater certainty of it to make Marianne's conviction of their attachment agreeable to her.
Elinor is a deeply emotional woman who puts the demands of sensible life over the urges of unrestrained sensibility.