Since Bitbucket is based on DVCSes, you won't actually lose any of your code even if they do go under, so it's probably a better question if you trust them with your issue tracking than with your code.
Sure, its DVCS, but if they go away, where will you host it now?
You are choosing to use Bitbucket for specific reasons. If they shut down because Atlassian is burning money and wants to trim the bottom line, then you now need to find something else to fill those specific needs again.
You don't lose any code, but you now need to figure something else out. And if you were relying on "free unlimited private repositories" you might not find that on another and now have to pay for something else.
Personally, I use GitHub daily for my job, but my private code isn't with GitHub.
On any available machine with some free space and an SSH daemon?
You talk about "figuring something else out" like it's some sort of big project. I don't understand that at all. There's no really effective lock-in here, users can take their repos and go anywhere or nowhere with negligible effort. They could even easily maintain a full copy of their repos on some backup machine at all times while using bitbucket or github.
