The Wi-Drive is one of the many Flash storage solutions offered by Kingston.As you can probably guess from the name, it has both USB and Wifi, and that means that you can access one by being on the same Wifi network. It comes in 32GB and 16GB flavors, and you can find them on Amazon and other websites.
Besides the KF app, there's also an app in iTunes and the Android Market.
This is a neat idea, but it's not your only option. I don't have a Kindle Fire, but I have looked at storage options before. First, Amazon already gave you 5GB of Cloud storage just because you are a US customer, and if you buy an mp3 album you can get another 15GB cheap (under a buck).
Wifi equipped drives are nothing new, so you could also look at Kingston's competitors and see what they offer. In fact, this tech is so well established that there are any number of generic external hard drive cases that also have Wifi and network support. You would have to access the files via a browser, not an app, but that might work better (fewer restrictions).
And then there are the various online storage sites like Dropbox, Box.com, ADrive, and others. Most offer a limited amount of free storage as a teaser in the hopes that you upgrade to a paid service. Some even have iOS and Android apps (Dropbox, for example).
And let me throw in one outside the box idea: calibre. One of its lesser known features is the ability to run on your PC and act as a content server over Wifi or other networks. It provides its own browser based interface, and you can use it to share your entire calibre collection. I've sometimes thought about using it to share files at a conference, but that's not possible these days (too much signal congestion), I also know of a number of people who use calibre on their home media servers as a central ebook library. It works well.