You can very quickly generate a lot of data with pictures of your kids. I have on the order of 80 GB with two kids under 5.
You definitely want multiple layers of protection, both locally and remote. For remote storage of pictures and videos, Flickr can't be beat price-wise. It is *unlimited* storage for $25 per year. And you can always retrieve the original file, and there are tons of APIs and clients available.
It's also useful for sharing photos and videos, with a strong security model that lets you control who has access to pictures of your kids.
Flickr does have a 500 MB per video file limit for uploads, and a 90 second limit for playback (you can download the original longer than 90 seconds, but no one else can view more than 90 seconds), but splitting videos up can be scripted with tools like ffmpeg, of course.
The key, though, is to *always* have more than one accessible copy of the originals in different physical locations. (i.e. two hard drives in your house doesn't count)
I also use an online backup solution. Look for unlimited storage for a reasonable price. I settled on CrashPlan+ Unlimited for $50/year, but there are a lot of options out there, now.