Can you sell by using paypal without a credit card or a bank account?
I started my own business selling clothes online so I thought paypal would be easy for customers to send their payment to. But I don’t want to use a credit card or a bank account to verify it. If I’m only using paypal to recieve money, I wouldn’t need a credit card/bank account, would I?
And then I can request a check every month?

Paypal handles all forms of payment including cards, echecks, bank, etc.
Where do you from? , myself mostly ask this services from mediator/broker who would to accept other payment methods my favorite is indonesia2world.com
In order to use your PayPal balance you have to have a credit card or bank account on file. You’ll need this information to withdraw the funds too. Otherwise the funds will sit in your balance and you won’t be able to do anything except refund them to the buyer.
For a business the best account is a bank account so you can easily withdraw your funds.