Don't the words, “odds of hanging on” conjure up an image of a paleontologist (you, of course) hanging off the side of a cliff, tool belt dragging you down as you struggle to free a fossilized rock from the sheer wall in front of you? That image sounds a lot more interesting, at least to the spectators, than the phrase's real meaning: your ability to keep your paleontological career going regardless of what chaos swirls around you.
Realistically, you'll have two factors to consider. First, you'll need the physical stamina and mental temperament to actually do the job. If you're a tenured professor with a nice cushy university office and a grad assistant or two, and if your department has a nice endowment that keeps funding your programs, you're basically set for the long term. Or you might be under contract to an oil conglomerate, which has luckily been doing very, very well for many years. You've got assistants who actually perform the grunt work, so you only have to tear yourself away from the golf course long enough to analyze the fossilized rocks your crew sends over. In both cases, you can probably hang on for quite a while.
Okay, let's say your job requires you to bust your butt every day; in other words, you're digging and chipping for fossils for hours on end. You also get your share of weather extremes, biting and stinging insects, and complaints from ticked-off students who had no idea field work actually involved the great outdoors. Here, your odds of hanging on depend on how long your body and sanity hold up.
You've also got one more variable to juggle: funding, or lack of it. Maybe you work for a research institute, a museum, or a university, and your department's budget always goes down to the wire. Maybe you work for an industrial firm that's on the skids, and the CPA wants to cut the fat so the company can stay solvent. In any case, your ability to hang on depends on your employer's ability to pay your salary. You might want to keep your ear to the ground, make sure your payroll checks clear, and always have a Plan B.