[I wrote most of this late last night and held onto it until this morning so I could look up a few things. In the meantime ftg has nicely responded to a number of my points.]
Here is the evidence that it’s illegal. It’s called “the copyright law.”
Here is the full text of the section on “fair use.”
Please show us where you find the exception for “personal use.”
I’ll find you that example after you find me a case of someone being successfully sued for defrauding someone of one dollar.
Please remember that copyright violations are, for the most part, handled in civil, not criminal, courts. So the fact that copyright holders choose to go after only the most egregious violators, where large sums of money are at stake, and not the millions of people who make personal copies, does not make those small violations legal, only too trivial for anyone to bother with.
I assume you are referring to the aforementioned Betamax case, in which time shifting of broadcast television shows for personal use was deemed permissible. IANAL, but I’ll remind you that that decision was made in a pre-digital, pre-Internet era. Once it was possible for anyone to make unlimited, perfect digital copies of music, movies, and other works, the laws were significantly updated.
One of the first such revisions was the Audio Home Recording Act of 1992, which Ruken brought up. But he admits that it probably applies only to audio recordings, and ISTM that there are legitimate historical reasons for that distinction.
Since you are the one making the claim that those cases apply by analogy to all other examples of copying for “personal use,” the burden is on you to show us the cases that support your argument.
I think you’ll have a hard time doing that, because copyright case law varies widely, depending on the medium involved. For instance, in print publications, the area in which I have had the most direct involvement, the question of what portion of the whole work was copied is a very big factor. I publish a print-only business newsletter, and have several times had to threaten people who were routinely making copies of the whole newsletter for others in their office. I’ve been advised that case law is quite clear that, under points 3 and 4 of the fair use section, copying a single article is generally fair use. But copying a whole publication is definitely not.
Here’s an article about one of the most infamous cases in my business: Lowry’s Reports vs Legg Mason. (Thomas Curley, the lawyer quoted in the article, represented me in some of the actions I had to take against infringers.) In print, the issue of copying is taken so seriously that the penalties can be up to $150,000 per infringement. Legg Mason bought one $700 subscription to Lowry’s investment newsletter, then faxed and copied it around many offices, including posting it on their internal intranet. They continued even after Lowry found out and demanded they stop. Legg Mason was found guilty and Lowry was awarded almost $20 million.
I think this case bears a much greater similarity to the OP’s situation than either video time shifting or making copies of MP3s.
Thank you, Justice BigT. Once the Senate confirms you, I’m sure you’ll have no trouble persuading at least four of the others. 
So as long as a lot of people do it, it’s legal?
Look, the copyright laws are far from perfect. If you want to argue that the period of copyright protection has been extended way too long, or that the DMCA needs to be reformed, I’ll stand right alongside you. But when you claim the law doesn’t say what it says, we part company.
Go ahead and make copies of movies, songs, and books without regard to whether the creators of that content have been properly compensated, and tell yourself it’s okay because it’s “personal use.” You almost certainly won’t get in any kind of trouble. Just don’t claim you have the law on your side.
TL;DR: What ftg said.