Sort of. The program — called "Share", as mentioned in the headline — is built on the same BitTorrent protocol we all know very, very well. The process of sharing isn't a stretch from what we're used to — all the new program does is provide an interface more suited to the purpose of storing and retrieving cloud-based files.
Is it designed to replace DropBox and similar services? BitTorrent says no, at least for the time being. In a comment to the original blog, the company revealed its expectations:
Right now, the product is geared toward sharing a file with a few friends, not long-term storage per se. However, as long as the file is still on your computer and in the group you’ve created, it’ll seed if one of your friends wants to download it.
So no exposing your hard drive to the internet masses — just those you know. According to reports (I couldn't find any references in the original blog post), Share uses Amazon's servers to handle the upload and initial seeding — so it's not like your files aren't going somewhere to start with.
It'll be interesting to see how it works — even with Amazon helping out (at what cost, we don't know) seeding seems a bit of a fuzzy way of ensuring your files are copied elsewhere. When it comes to storage, I prefer it to be a binary operation — either my file is stored in certain place, or it's not. I can't see this being a great way to making reliable offsite backups, for example.
BitTorrent's made it clear the program is in alpha, so I'm sure it's using the time and testing to judge just how practical and useful the software will be and how much it needs to scale. Everyone in the world owning a bit of everyone else's files? That's an interesting future, for sure.