If my grandfather passed away and was living with his girlfriend. Am I able to obtain his possessions? They were together 4.5 years at her residence

Assuming that the property was owned by your grandfather (i.e. that he had not gifted it to her, it was not jointly owned, etc.), it would pass according to his will, which might have to be probated, particularly if there are any disputes as to anyone's rights to it.  If your grandfather had no will, his property would pass according to the laws of intestacy.  I am not familiar with the intestacy laws of D.C., but in general intestacy laws don't protect girlfriends; property would be divided amongst close surviving relatives.  Thus, for example, if your grandfather was survived by any children, grandchildren may not receive anything.