Abraham's bosom is neither a physical location nor simply an embracing relationship, but a spiritual realm. A note in the NABRE explains: "Abraham’s bosom; or lap [is] ancient Jewish terminology for the place of the righteous dead."
To summarize, in the period between the closing of the Old Testament and the advent of Jesus, the Jewish concept of the afterlife was evolving and a considerable literature about this has survived. The concept of Sheol -- a place where souls lived in a kind of stupor -- gave way to a belief in several differentiated realms. Paradise was the best of these, Gehenna (also called Hades or Hell) the worst. In this literature Abraham was a central figure. He was sometimes depicted as a gatekeeper to either Paradise or Gehenna, or both. To be embraced in his bosom was to dwell the highest realm.
According to the Testament of Abraham (first century c.e.) God instructs the archangel Michael:
"Take, then, my friend Abraham into Paradise, where there are the
tents of my righteous ones and (where) the mansions of my holy ones,
Isaac and Jacob, are in his bosom, where there is no toil, no grief,
no moaning, but peace and exultation and endless life" (T. Abr. 20:14)
In the Jewish Encyclopedia article on this subject, the data are analyzed. Its author, Kauffmann Kohler, says of Jesus' story:
Abraham is here viewed as the warden of paradise, like Michael in
Jewish and St. Peter in Christian folk-lore.
So, to be in Abraham's Bosom is to be in the Paradise, the highest realm of the spiritual realm available to humans, according the view of 1st century Judaism.