If a business only takes cards and not cash, like a hotel, is that illegal?

In the 21st century, that is a very good and complex question. In most developed jurisdictions, legal tender means that the designated money is valid payment for all debts unless there is a specific agreement to the contrary. So that, for instance, if you go into a cafe that has a notice "NO CASH PAYMENTS ACCEPTED, and you order coffee, you are implicitly accepting that as part of your contract.

But where there is no such understanding, then generally it would be illegal in most jurisdictions.

But its a question that technology makes more complex every year!