We think that recording streaming videos for later personal enjoyment or edification of yourself and members of your immediate household is allowable, whether as a "fair use" under the United States Copyright Act or the equivalent in the territory of the rights holder or you, otherwise, it is the equivalent of using a digital video recorder (DVR), or an old fashioned VHS or Sony Betamax video recorder (we suspect that most of you don't remember the Betamax) hooked up to your TV, or a video camera pointed at your screen, to record.
And there are many videos that are in the public domain (meaning that, at least in the United States, there is no copyright or all copyright has fully and irrevocably expired), or where the copyright holders are okay with you saving or recording these videos for your later personal enjoyment or information.