The JPS Tanakh suggests "base of the mountains" as a translation for the phrase you indicate.
The unvocalized Hebrew is קצב (qṣb), which the Masoretic Text interpolates as קֶ֫צֶב (qě·ṣěḇ), meaning "shape" or "foundation" (Dictionary of Biblical Languages with Semantic Domains) - consistent with the Tanakh translation and the YLT version you quote.
An alternate interpolation would be קָצַב (qā·ṣǎḇ), which according to the same source means to "cut off", but this is not, I think, what is actually in the Masoretic Text.
Interestingly, the Greek Septuagint version of this passage (which is older than the Masoretic Text) reads:
εἰς σχισμὰς ὀρέων, κατέβην εἰς γῆν ἧς οἱ μοχλοὶ αὐτῆς κάτοχοι αἰώνιοι·
καὶ ἀναβήτω φθορὰ ζωῆς μου, Κύριε ὁ θεός μου
To the clefts of the mountains; I went down into the earth, whose bars are the everlasting barriers: yet, O Lord my God, let my ruined
life be restored.
(Brenton's English translation) which seems to agree more with qā·ṣǎḇ than with qě·ṣěḇ.
If I had to guess, I would say that the NASB translators observed the discrepancy between the Masoretic Text and the Septuagint, looked for a basis for the Septuagint translation, and concluded that קָצַב was more faithful to the original Hebrew than what was chosen in the Masoretic Text.
There are quite a few very sharp Hebrew scholars on this site, so I am sure that if my conclusion is weak they will be all over this answer soon.
As for why Jonah says "forever", I assume it was because at the point that he was praying what is in the Bible text, he did not know when or even if he would be released.