Users grant StackExchange a licence:
You agree that all Subscriber Content that You contribute to the Network is perpetually and irrevocably licensed to Stack Exchange under the Creative Commons Attribution Share Alike license.
That should be determinative, but there was also some question as to whether StackExchange users can be deemed to have agreed to these terms. For this reason, I'll also review some case law relating to what are known as "click-wrap" agreements where the terms are made available via a hyperlink.
In my opinion, StackExchange's way of displaying links to their Terms of Service during registration meets the requirements that have in the past been sufficient for courts to deem the user to have read and agreed to those Terms of Service. See for example, Schnabel v. Trilegiant Corp., 697 F. 3d 110 (2012), especially the section titled "Notice" for reference to other cases:
A person can assent to terms even if he or she does not actually read them, but the "offer [must nonetheless] make clear to [a reasonable] consumer" both that terms are being presented and that they can be adopted through the conduct that the offeror alleges constituted assent.
Specht v. Netscape Communications Corp., 306 F. 3d 17 (2002)1 frames the notice test in terms of a "reasonably prudent offeree" and whether they would "have known of the existence of license terms".
In Guadagno v. E Trade Bank*, 592 F. Supp. 2d 1263 (2008), the court found that clicking on an acknowledgement icon near an underlined, highlighted link to an agreement was acceptance of that agreement:
In the instant case, a highlighted, underlined link to the Agreement was directly above the acknowledgement box, along with notice that "The following contain important information about your account(s)." A reasonably prudent offeree would have noticed the link and reviewed the terms before clicking on the acknowledgment icon.
I think the most similar case is Fteja v. Facebook, Inc., 841 F. Supp. 2d 829 Dist. Court, SD New York (2012), although not at an appellate level.
In order to have obtained a Facebook account, Fteja must have clicked the second "Sign Up" button. Accordingly, if the phrase that appears below that button is given effect, when Fteja clicked "Sign Up," he "indicat[ed] that [he] ha[d] read and agree[d] to the Terms of Policy."
This is very similar to StackExchange's sign-up process:
A DMCA takedown could be successful if submitted by somebody other than the StackExchange user where that other party asserts copyright ownership in the contributed material. This could happen if a StackExchange user infringed copyright by posting material that they didn't have the right to reproduce.
1. Opinion authored by now Supreme Court Justice Sotomayor.