What is the Biblical basis that God's sacrifice is sufficient for all of our sins (past included)? I spoke to a friend recently and they said:


  I feel like I've done so much in my past, and that is holding me back from having a relationship with God. How can I know that God will accept me, even after I have done wrong in my past? Can God really look beyond my past?


My question is probably best asked as:


What is the biblical basis to support the position that a person's sins (past included) do not preclude them from being able to receive salvation?


I'm looking for answers that come from Bible passages (purely exegesis). My friend doesn't subscribe to a specific denomination; with that in mind, I'm searching for Biblical answers rather than Catholic or JW doctrinal backed answers, for instance.
The question you are asking here is really the essence and the magnificence of the Gospel itself. The scripture that we're all familiar with John 3:16-17


  16 “For God loved the world so much that he gave his one and only
  Son, so that everyone who believes in him will not perish but have
  eternal life.
  
  17 God sent his Son into the world not to judge the
  world, but to save the world through him.


Jesus knew what your friend has done when he died on the cross and died for him just the same.

If you are looking for precedence see these examples:


The Samaritan woman in John 4 - she was not living a moral life yet Jesus takes time to reach out to her.
The Woman caught in adultery in John 8 
Peter, Jesus' disciple denying him in John 18 and then Jesus restoring him in John 21
Paul in Acts 8 and 9 - he killed many Christians
The ultimate example comes in Luke 23:34 when Jesus is on the cross and He forgives the very ones crucifying him.


Jesus even talked about this in a few of His parables for example the lost sheep and the prodigal son.