The author is completely saturated by disasters; he is filled to the brim with them. His life approaches death. The ESV does not translate the word Sheol.
Yet, there are good reasons to describe this as the realm of the dead.
This was understood as the place where the dead disappeared completely from sight. All hope of life was futile there. Heman is afraid of that place. That is why he fervently calls upon the Lord. Three times (in Psalm 88:1, Psalm 88:9, and Psalm 88:13) he cries to God to save him from (being in) that dreadful realm of death.
3 For my soul is full of troubles, and my life draws near to Sheol.