First, with regard to what kriskros54 and Carling said:
kriskros54, it is safer to give out your credit card number to a merchant/company like PayPal than it is to give them your checking account number. The laws and regulations are different for each, and it's MUCH easier to dispute a credit card charge than a direct checking debit, whether through a debit card or direct ACH transfer.
Carling, I do see your point, like maybe the questioner had gone to a phishing site, but I think he is just talking about the normal PayPal procedures—they DO require either a checking account or that you sign up for their own credit card, in order to be a "verified" user. You can still do certain things without being verified, most people wind up wanting to have the verified status.
OK, now for my answer:
Please see http://www.paypalsucks.com/. Don't get me wrong, I use PayPal and EBay myself. However, given what that site talks about and knowing in general how shady PayPal and EBay seem (to my taste and based on my reading) to be, I would do as follows:
1. Open a separate checking account that you will use ONLY for PayPal.
2. Give PayPal the info for that checking account as well as info for a credit card (if you want). (They require the checking account in order to be verified, unless you sign up for their PayPal Plus Credit Card, which I would not do unless you really want it for some reason.)
3. I recommend that you DO NOT give PayPal a debit card number, but if you do then give them one which is ONLY linked to the account that you set up in Step 1.
4. As needed, transfer funds into the account you set up in Step 1 to cover your PayPal transactions. Or, as one answerer pointed out, just use your credit card to transfer any funds as needed, and keep that bank account empty or very close to it, and only use it to collect the money FROM PayPal when your PayPal balance gets high enough.
Why all of this? It sounds complicated, but it really isn't, and it will protect your money reasonably well. Your existing bank should be able to set up a second account for you, one with no fees, no minimum balance, and which pays you no interest. By using the above methods, you can have the benefits of having a "verified" PayPal account; but you can just keep a very low balance in the account that is linked to PayPal—for me that is $50. For you maybe it will be $1,000 and for someone else maybe it will be $1. In any case, this way if PayPal or EBay screws something up, they will only be able to tie up $50 of my money (unless your bank is stupid and transfers the money from your other account.) Also, I think it's safe to give them a credit card number, because if they make fraudulent charges against your credit card you can easily dispute them. Plus, a credit card charge ties up the card issuer's money—not yours. The credit card cannot legally hold you liable for fraudulent charges. Let PayPal fight your credit card company; your credit card company will have a lot easier time winning against PayPal than you will. In contrast, if you give PayPal or anyone else the ability to directly debit money from your checking account, that money is MUCH harder to get back, even if it was taken fraudulently. You would in all probability get the money back eventually, but think of the consequences of not having the use of $100, $1000, or $10,000 (whatever amount they screw up and incorrectly take from your account) for some period of months while you figure it out legally. With a credit card, you are legally required to pay only the first $50 of any loss due to theft/fraud, no such limitations exist with a checking account. [Note: This is all assuming you are a U.S. citizen.]
Full Disclosure: I am not affiliated with the paypalsucks.com Web site, and I also have not had any negative experiences with PayPal or EBay in this way. This is just how I personally handle my interactions with them for the sake of caution.
Hope this info helps you.