EDIT April 2016
Actually, I have recently been reading in depth about LGPL, and asking some experts on the subject. Turns out that the inability to use static linking for Qt in closed source applications is more of a cultivated legend and has nothing to do with reality.
What LGPL requires is the possibility for the end user to relink the application against a different version of the library. Using dynamic linking is one way to achieve that, but you can just as easily provide your object files, this way you don't have to open your source and the LGPL requirement is still satisfied.
On the Qt website there are 2 legal FAQs and in neither of them is it directly stated that you can't do it. Just as it is not stated that you can. However, there are at least several instances of implying a vague legal threat in case that you do. I think there is a good explanation for all of those - they can't say that you can't do it without publishing a practical lie which may as well have negative legal repercussions for them, and they are willing to discourage doing that rather than encourage it, as it has the potential to force more people into purchasing a commercial license.
So in short, yes you can, and you most definitely should as lately Qt has become a living deployment hell, plus as of Qt 5.7 in a static build, QML files are tucked neatly in the executable rather than being out on the file system, for anyone to tamper with. Just make sure that:
- your Qt build only contains modules, licensed under LGPL, and nothing GPL
- the about section of your app mentions that it is using Qt and contains a link to where you can download the application's object files
- include all the respective license files with your application
Lastly, your application actually has to be "relinkable", that is, it must be able to work with a compatible library version that provides the necessary functionality. Which means that if you have made modifications to Qt before building it, you must provide those in the form of source code as well, but only the modifications to Qt, not your application's source code.
Update:
Here is an excerpt directly from the GNU FAQ:
For the purpose of complying with the LGPL (any extant version: v2,
v2.1 or v3):
(1) If you statically link against an LGPL'd library, you must also provide your application in an object (not necessarily source)
format, so that a user has the opportunity to modify the library and
relink the application.
That states it pretty clear.
The old, original answer:
It is possible to build Qt statically as long as your application is open-source and you provide the source. If you want to keep your source closed, you either need an expensive commercial license, or you need to use dynamic linking.
BTW using a static build of Qt is pretty nice, for Qt5 I get about 7-8 MB executable with no external dependencies, which is much better than the 20+ MB of additional dll's you have to ship with a dynamically linked app.
For more information, you can take a look at this video: Making the correct license choice when developing with Qt
All in all, can it be done? 100% yes. Should it be done? It depends, for personal/testing/learning purposes it is 100% OK, however if you plan to distribute production grade software, be that commercially or not, open source or not, you'd better first consult with a lawyer. The whole subject is unnecessarily complicated, subject to interpretation, so that the consultation with a lawyer becomes more expensive than a pricey commercial license.