Would people be able to survive indefinitely with radiation and nuclear winter at other places not directly hit?
Holocaust would mean to any extent of operation and damage and not just limited to a single targeted territory and so the extent of devastation would not be gauged just by a simple question and answering in the way others are attempting to! Any nuclear war, if we foresee a one, would never be a limited war game and would stretch the world over, with a fallout of a nuclear winter, just darkness for not less than ten thousand years as per the lowest estimates and would wipe out all that on earth and nothing would have a surviving possibility, and by no plants to grow for a hundred centuries the life forms would all vanish, leaving the nature to orient a plan of its own to generate the life forms as species by evolution once again and to make men and women like us, perhaps in just another hundred millions years from now!
There's no way to accurately answer this question. Could people go underground and survive until the radiation fallout dissipated? Yes. In fact, there are shelters that are capable of allowing people to survive out nuclear war as we speak. Good luck finding one, of course. I'm sure they're mostly reserved for scientists, doctors, and such :) I believe we would survive, but the world as we know it would be gone. The majority of life could potentially be wiped out. Fortunately, we can currently survive on current technology, without any earth resources. They do it on space stations all the time, and have grown plants and food without sunlight for decades. Could we survive? All I can say is, adaptation can be amazing. The question of survival isn't about if we would all be killed by the holocaust or not, but if we could as a species adapt to the changes, and continue forth. I say yes. Some would survive. At least, for a while.
The human race and other species would be decimated, but I think humans are more clever and adaptable than some might think. A nuclear winter would generate strong winds, which could turn turbines, which could provide the light needed to grow food underground. Uncontaminated water would have to be recycled as has been done in experimental bio-domes, as would air, but it could be done. Some people would also make it to Mars or space stations if the nuclear holocaust doesn't happen too soon. None of that lessens the tragedy that would result from a nuclear war.
The human race is very adaptable, and inventive. Some people would survive, and in the end, I don't think anything short of the total destruction of the planet (sun going nova, planet literally shattered by a massive metor hit). We couldn't do it if every nuke we had got launched. We are as tough as roaches when you get right down to it. Trust me, no matter how many nukes went off, there would be survivors, and they would scrape by.
I've read that those in deep silos, isolated canyons, and south Pacific Islands away from the tradewinds....all with uncontaminated water supply, could survive a nuclear winter. The deep silos would need provisions for two years before venturing outside. Of course this is all theoretical.