It is not specifically stated in the books that Ariana Dumbledore was an Obscurial, but the description Aberforth provided to Harry about her fits the description of an Obscurial perfectly.
"It destroyed her, what they did: She was never right again. She wouldn't use magic, but she couldn't get rid of it; it turned inward and drove her mad, it exploded out of her when she couldn't control it, and at times she was strange and dangerous. But mostly she was sweet and scared and harmless."
—Aberforth Dumbledore, Harry Potter and The Deathly Hallows
Description of an Obscurial:
When a magical child is forced to repress their talent through physical or psychological abuse, they develop a parasitical magical force inside of them called an Obscurus, resulting from their strong emotions of distress.
An Obscurial may lose control when they reach their emotional and mental breaking point, releasing their Obscurus as an invisible (or nearly invisible) destructive wind.
Ariana (possibly) became an Obscurial due to the attack by some Muggle boys who saw her practising magic when she was six years old. This led her to suppressing her magic and (possibly) developing an Obscurus.
This also leads to other questions which I shouldn't answer as no one asked them, but I will anyway, because you can never have too much information.
Why wasn't Harry Potter an Obscurial?
He seemed like the perfect candidate. Emotionally and mentally abused by his relatives because of what he was.
On her official website F.A.Q., author J. K. Rowling answered: "An Obscurus is developed under very specific conditions: trauma associated with the use of magic, internalised hatred of one’s own magic and a conscious attempt to suppress it. The Dursleys were too frightened of magic ever to acknowledge its existence to Harry. While Vernon and Petunia had a confused hope that if they were nasty enough to Harry his strange abilities might somehow evaporate, they never taught him to be ashamed or afraid of magic. Even when he was scolded for ‘making things happen’, he didn’t make any attempt to suppress his true nature, nor did he ever imagine that he had the power to do so."
But anyway, back to Ariana.
Yes, it is entirely possible that she was an Obscurial and maybe the reason why J.K. Rowling hasn't confirmed this is because it has something to do with the plot of the next Fantastic Beasts.
In fact, the only real evidence against this theory is how old Ariana was when she died. 14. Obscurials generally live to the age of ten before the Obscurus bursts out of them, killing them in the process, but, thanks to Fantastic Beasts, we now know that this isn't always the case in the form of Credence Barebone.
Speaking of her dying, we don't quite know how she died except that it was in a duel between Albus Dumbledore and Gellert Grindelwald. The description I could find was:
A three-way duel broke out between him, Albus, and Aberforth. Ariana, aged about 14 at the time and unable to control her magic, tried to stop it, and was inadvertently killed; it was never known which of the three wizards was directly responsible for her death.”
It is plausible that the magic she could not control was due to the Obscurus inside her, which knocked the two boys of course as they cast the killing curse. The curse, instead of hitting the other, hit Ariana instead, and she was killed.
So, in conclusion, it is highly likely that Ariana was an Obscurial.