Whenever I've accurately felt this way - not simply feeling overawed by the height of mountain to be climbed - it has typically indicated: 1) lack of empowerment of IT teams to push back on features and deadlines; 2) poor grasp by business stakeholders of availability and ease of reallocation of IT assets; 3) unrealistic goals by project owners or managers.
I no longer accept impossible terms. If I identify factors that create a seeming "impossibility state," I offer the managers and owners options to restore a state of non-impossibility, e.g., more time, more money, fewer features. Replacing me, the troublesome doomsayer, is an option...prob'ly won't fix the impossibility issue, but, hey, no more of that bothersomely distracting, truthy stuff. If I present an impossibility finding and it's rejected without CONVINCING counter-argument, I won't continue without an explicit waiver of liability.
I've been programming, analyzing, architecting, and engineering software and data solutions for over three decades. I'm a mathematician with formal education and professional experience in the methods of Advanced Statistics, Operations Research, Decision Theory, and Project Planning. I don't casually say "impossible" - if I say it, I can back up my claim with non-trivial arguments.
I can help tame the impossible in shops that aren't crazy, and I won't work for a crazy shop more than once. However, I only seek part-time and temporary positions these days, and hence, I'm less locked-in than most captive employees or even small business owners (my bride's a working professional - I eat and sleep indoors, whether I work or not).
TL;DR: Yes, I've labored on "Impossible" projects - will NOT do it again. Hard? Fine. Impossible? Nope.