The expression 'distant sky' appears in old poetry.
遠い is the plain modern word for physical or temporal distance, while 悠 evokes a vast, boundless distance in archaic or poetic style.
落ち着いた is the common modern way to describe a composed manner, whereas 悠 conveys the same composure but only in old or literary usage.
果てしない is the standard modern adjective for 'endless', matching the boundless-in-time-or-space sense that 悠 expresses archaically.
The kanji 悠 combines 心 (heart) and a phonetic component, originally meaning 'distant' or 'long-lasting'. The exact historical derivation of the Japanese reading ゆう is uncertain.