It depends on what country you reside. In the USA, UK, Israel etc, you have a fair use clause and you might be allowed of doing it.
In other countries, alone the fact that you are storing the content (without publishing it) is blatant theft.
In other countries, alone the fact that you are storing the content (without publishing it) is blatant theft.
That being said, a DMCA notice on the site would protect the content provider. The way it works is that if you have a copyright claim against the site, they have a duty to take it down when you inform them. As long as they respect that, they are pretty much covered.