To be up to one's neck (in debt or work); to be overwhelmed and unable to manage finances or tasks.
投資に失敗して首が回らないほど借金を抱えてしまった。
I failed in an investment and ended up with debts so big I can't keep my head above water.
年度末で仕事が山積みで、今はとても首が回らない。
With year-end piling up, I'm completely swamped right now.
支払いが重なって、今は首が回らない状況だ。
Payments have overlapped; I'm in a situation where I can't make ends meet right now.
Literally, to be unable to turn one's head — i.e., physical inability to rotate the neck.
寝違えて首が回らないから、今日は運転できない。
I slept wrong and can't turn my neck, so I can't drive today.
事故で首が回らない症状が出たため、すぐに病院へ行った。
Because I developed an inability to turn my neck after the accident, I went to the hospital immediately.