If my father passed away over 2 years ago and my brother and I want to access some of his belongings, how do we go about it without using a lawyer?

If your father passed away without a will, then the state laws of intestacy will apply to administration and distribution of his estate.  If you and your brother are the only heirs (i.e. your father did not leave a spouse or other children), you should start by filing a Petition for Probate with the superior court of the county in which your father resided at the time of his death.  In the petition, you should ask the court to probate your deceased father's assets without a will, and to appoint either you or your brother an administrator of estate.  

Once the Petition for Probate is filed with the court, you have to follow all the procedural formalities (notifying other potential heirs about the date of the probate hearing, publication in the local newspaper etc., depending on the laws of the state), attend a probate hearing, and have the court issue Letters of Administration.  This document will allow you to access your deceased father's assets and begin the process of administering and distributing his estate.  

Although you can go through the probate process without an attorney, the procedure is fairly complex, and mistakes can be costly.  I strongly suggest you consult with an attorney knowledgeable in estate adminsitration and probate.