Alarmist article. Doesn't mean Flickr is going to disappear any time soon, nor Yahoo!
Yahoo! may not be doing as well as some of the competition but they are not going away soon.
Yes there are loads of Yahoo! services that die, but that's because they are crap. They keep coming up with new ideas, and they're crap, or there's a better alternative that everyone uses. For all the services that the article mentions have been shut down, Yahoo! have kicked off new ones (though most are still crap).
However Flickr is not a service Yahoo! invented. They bought it. It was great, and still is, and it beats all the competition for social photography. It's way too popular to kill off, and on top of that they make money from it.
Their core search service is not king of search, but it's still widely used and again they make money from the advertising.
There is a risk that Yahoo! goes tits up, but that doesn't mean Flickr goes with it. Almost certainly Google, Microsoft, or maybe even Getty, will buy Flickr from Yahoo! or even just buy the whole of Yahoo!
I'd say Flickr is sufficiently successful that it could be spun off as it's own business and survive on its own.
But yes, regardless, I wouldn't rely on any online photo sharing service as a backup.