BTDT, I've found two ways to handle it that have proven to work very well. I don't approve of these tactics ordinarily, but someone dumped a no-win situation on you and set you up to fail. That calls for drastic measures.
First, delegation is your friend. Dump as much responsibility on unsuspecting dupes as you can. First and second year engineers are easy marks. If you can drag in two or three interns you are on easy street.
Second, build a kingdom. You need resources? Go get them. you need people, offices, meeting rooms, lunches, outside contractors, consultants, etc.
You need to "borrow" as many people as you can, just get them assigned to you as a temporary assignment.
Make darn sure you focus on the org chart and big picture, don't get tied down in the details.
Turn this project into the biggest, most expensive, and resource consuming cluster _uck you can but always focus on the big picture and the presentation. Glossy covered books, manuals, websites, whatever you can do to make it appear to be a world class process.
It will drive your direct supervisors crazy and they will try to figure out how to get you off that project without drawing any attention to it, but you will impress the heck out of the people above your supervisor one or two levels who don't have a clue what is really going on.
You will eventually be pulled off the nightmare project and will get a better assignment. Sometimes they will create one just for you.
This is very risky but if you know how to play the game you will come out ahead.
One other thing, be prepared to change jobs in 4 or 5 years, if you stay at one place too long eventually it will come back to haunt you
Yes this is a bit of tongue-in-cheek but there is also some truth in it.