Is it possible for my husband to receive his residency within the U.S.?

Under the current immigration laws, he is not eligible to legalize his status to permanent residence within the U.S.

No. He has to first obtain an unlawful presence waiver. If granted, he will have to return to his home country and consular process back into the US. There is no other way.

If he has no criminal record and was never removed from the U.S. by the Immigration, there is a procedure that can give him a good chance of becoming legal. Consult an immigration attorney to see if a case can be started with a reasonable chance of success.

Yes, it is possible for you to help your husband get his residency. However, he will need to return to his home country in order to apply for his immigrant visa (permanent residency) at the US consulate. Assuming you are a US citizen, the first thing you need to file is an I-130 relative petition on his behalf. After the petition has been approved, he will need to apply for a provisional waiver on form I-601A because he has been unlawfully present in the US in excess of 1 year. If the provisional waiver application is approved, he should have no problems returning to the US after his immigrant visa has been granted at the consulate.