It might not have been a technical glitch with Flickr. They might just have changed their site structure - you can't rely on a free third party service to guarantee a 'permalink' type structure in their site.

Your question doesn't seem to relate to backups as such, I presume you've got some sort of backup system in place already, and that you are exclusively asking about an easy way to share images.

The best way to be sure that a link you use isn't going to break is to purchase your own hosting. This is the only way you can really guarantee anything (excepting the fact that you are dependent on your hosting service not having issues). Paid hosting with your own domain is relatively cheap. You can use various off-the-shelf and open source gallery type sites, Wordpress, Joomla or any other content management system. Obviously the trade-off is that it's up to you to configure it and maintain it. If you're prepared to spend a bit of time though, a simple self-hosted Wordpress site may be worth creating.

This isn't a free solution, but it doesn't have to be expensive, but at the end of the day, you need to have some sort of solution that you have control over.