You're not wrong if you think being in a pickle sounds old-fashioned. The Oxford English Dictionary (OED) has the earliest-known written instance of this phrase as from 1562. Granted, it's early days for modern English, so it looks a little odd to us now, but the idea is there:
"Man is brickell. Freilties pickell. Poudreth mickell, Seasonyng lickell."
In this quote, the author John Heywood is saying that man is brittle, or frail, and that frailties pickle, or keep, like pickled foods do. Even the OED admits that this early example is a little weak. But in this quote from 1585 by John Fox in a sermon he gave on 2 Corinthians, the meaning is closer to how we use it today:
In this pickle lyeth man by nature, that is, all wee that be Adams children.
By nature, humans are in a pickle, which certainly feels correct most days. By 1711, we can easily identify how the author Richard Steele was feeling:
I am ashamed to be caught in this Pickle.