Do we deserve eternal life?

Do we deserve eternal life? Are we worthy of salvation? True, we cannot earn our way to heaven, but does that mean we are not worthy? Did Jesus die for us because we are unworthy? Is all of heaven sacrificing their time and talent for us because we don't deserve it?

Is unworthy the same as worthless? Is unworthy the opposite of worthy? What does the Bible mean when it says -

Matthew
10:37 He that loveth father or mother more than me is not worthy of me: and he that loveth son or daughter more than me is not worthy of me.
10:38 And he that taketh not his cross, and followeth after me, is not worthy of me.

Acts
5:41 And they departed from the presence of the council, rejoicing that they were counted worthy to suffer shame for his name.

Ephesians
4:1 I therefore, the prisoner of the Lord, beseech you that ye walk worthy of the vocation wherewith ye are called,

Colossians
1:10 That ye might walk worthy of the Lord unto all pleasing, being fruitful in every good work, and increasing in the knowledge of God;

2 Thessalonians
1:5 [Which is] a manifest token of the righteous judgment of God, that ye may be counted worthy of the kingdom of God, for which ye also suffer:
1:11 Wherefore also we pray always for you, that our God would count you worthy of [this] calling, and fulfil all the good pleasure of [his] goodness, and the work of faith with power:

Must we go around feeling unworthy as if we are sinful and worthless? As we grow closer to Jesus in friendship and likeness will we feel more sinful and unworthy? What does Paul mean when he wrote -

1 Timothy
6:6 But godliness with contentment is great gain.

As we become more and more like Jesus will we feel more and more sinful and unworthy?