Yea actually I think you can have one without a credit card, but not without a bank account.. My paypal was in my mom's name because we used her credit card. And for the account to become "verified" she needed to have a bank account with it too. Oh actually, I'm lying. She only needed to use the bank account to transfer us the money.
No, you can sign up and have an account with money in it without having a credit card, bank account, etc. However, you need to verify the account with a bank account or credit card in order to actually use the money inside.