Can you still get a green card if you entered the country without a visa?

No unless you are less than 30 years of age and entered US before you were 15 years old.Making use of deferred action for childhood arrivals (DACA) policy you can file Form I-821 D in order to legalize stay in US.You need to be a graduate or a GED certificate holder with clean records and proof to prove that you are eligible to benefit from this policy and possibly become a green card holder later!