You probably have about a hundred trillion different microorganisms living in your gut right now, which is ten times the number of cells in your body. These microorganisms help keep you alive, but researchers have figured out how to get them to store your porn collection at the same time.
Bacteria have the potential to store a lot of data. One single gram of e.coli could theoretically hold something like a million gigabytes of information, while one gram of the stuff that your computer's hard drive is made of can store about four gigs, if you're lucky. The way that the data gets stored is basically the same, though: your hard drive stores data magnetically by converting them into zeros and ones, while bacteria store data chemically by converting them into nucleotides and making DNA.
The problem with storing DNA data in bacteria is that there's a physical limit to the amount of data that each DNA strand (and each bacterium) can hold. The solution, of course, is to chop your data up into lots of little pieces of DNA, and give each piece to a different individual bacterium. When you do this, you also have to give each piece an address of sorts, so that you'll know how to put all the random pieces back together the right way. This sounds like a pain to have to do, and it is, but it has the side effect of encrypting your data pretty well, since without the address key, there's no way to put all the DNA snippets back in the right order.
Once you've got your DNA-encoded data inside your bacteria, it doesn't bother them in the least. They'll just go about their happy little bacteria lives, oblivious to the fact that they're being hijacked as hard drives. And when they reproduce, they'll duplicate your data at the same time, providing a massive amount of redundancy.
So now that you've got a couple million bacteria wandering around with all of your data in their tummies, how do you actually get it back out again? It's not so hard, as long as you have a fancy next-generation high-throughput DNA sequencer. The poor e.coli who have been loyally storing your data for you get all squished up and run through a machine that can read their DNA, and it spits out a big long list of all of those individually addressed pieces. Put them back together, and there you go, it's your data.
Now, the important question: what, exactly, are the chances of this random DNA causing the e.coli to mutate into some superbacteria that will destroy all life as we know it? Apparently, pretty low. Not nonexistent, mind you, but it most likely won't happen.