Not likely that you will be able to re-open any account that had a negative balance. Since this is 3 or 4 years old, Paypal probably charge off the account and sold the debt to a collection agency.
You MIGHT be able to open a new Paypal account, depending on how long they keep records of defaulted accounts. If they have record of you previously defaulting, they may not permit you to open a new account.
Personally, I don't know why anyone would bother with a Paypal account. They require direct access to your bank account and a credit card. Then charge you all sorts of fees. There also seems to be a whole lot of problems over disputed charges.