Definition
The Greek word χαρακτηρ denotes "an engraving" (source).
Translation
The modern day translation "imprint" of ESV perfectly corresponds to this meaning.
The translation "exact likeness" is actually not a literal translation but a dynamic equivalence.
Exegesis
The Lord Jesus is a copy of God's being. It means that Jesus is not the original being but a mere copy of another's being.
However, copies contain everything the original contains. Therefore, the Lord Jesus owns everything God owns.
The ESV translation has faithfully preserved the meaning of the Greek in contemporary 21st century English:
"...exact imprint of his nature..." [χαρακτηρ της υποστασεως αυτου]
Application
The Nicene Creed was made in 325 C.E. in order to respond to the new teaching of Arius. The Homoousian versus Anomoian debate has ended and the former has won the lengthy discourse. Hebrews 1:3 is one of the proof-text of the Homoousian party. It supported the Greek word homoousios (of same substance) to be used in the Creed (source).
Q & A: In Hebrews 1:3 does χαρακτηρ της υποστασεως αυτου say that Jesus is the same substance or similar substance to God?
The interpretation that Jesus is of similar substance with the Father is open to Hebrews 1:3 when read in isolation but since other NT texts already spoke of ontological oneness (sameness) between the Father and the Son [John 10:28-30; Col. 2:9], we ought to interpret Hebrews 1:3 in that light.
On the other hand, as already explained earlier in this answer, being a copy only makes you distinct but not different from the original since copies always carry all of what the original carries. Therefore, the phrase χαρακτηρ της υποστασεως in Hebrews 1:3 says that Jesus is of same substance with God.