Not permanently because all radioactive elements decay. So - can it be done temporarily? Well, not using modern nuclear weapons. Forget all those post apocalyptic stories of a radioactive world - after a real nuclear attack most of the radiation would be gone within hours, through days to weeks. After all, Hiroshima and Nagasaki are both thriving modern cities. Local hot spots could last for longer period, perhaps decades, so you would be well advised to have a Geiger counter, or more likely, they get cordoned off. None of that could make the whole world uninhabitable to humans.

Instead you have to look for a weapon specifically designed for this job. Who would be crazy enough to do that? Well not as an actual weapon but as a thought exercise, yes, someone has tried to invent such a bomb.

Theoretically if you could dust the entire Earth with cobalt 60 using cobalt bombs you could make it uninhabitable to humans - except of course in protected areas - for instance inside domes covering areas cleared of the cobalt 60. The idea originated with Leo Szilard - not intended as a serious proposal for a nuclear bomb, but as a way to show theoretically that humans could in the near future have the capability to design a "Doomsday weapon".

Physicist Leo Szilard who first hypothesized the nuclear chain reaction, and patented the idea of a nuclear reactor.

He devised the theoretical idea of a "Cobalt Bomb". The bomb contains a large amount of ordinary cobalt and the neutrons from the blast turn it into dangerous cobalt 60. It's designed to produce less radiation damage in the first few hours, but more radiation damage over time periods of years and decades. Areas affected would only become habitable to humans a century after the impact. It was just a "thought experiment" to show that we could in principle build doomsday bombs. You’d only need a few hundred kilograms of cobalt.

The idea is that cobalt 60 has a long enough half life so that it could spread throughout the Earth's ecosystems before it decays away enough so that it no longer kills people.

In practice it would probably still have a patchy distribution. It’s hard to see how they could explode enough bombs to make sure that every part of the Earth’s surface is dusted with cobalt 60.

Anyway we don't need to worry about cobalt bombs. Two nuclear tests have produced cobalt 60, but not by design. In one case the cobalt came from pellets used as tracers, and in the other case from the steel casing. For more on this see the wikipedia entry on Cobalt Bomb which has lots of cites to follow up.

This is perhaps the closest we have to a doomsday scenario but no such bombs exist. After all it's not to anyone's advantage to make the world uninhabitable. I suppose if you want to fix the nuclear doomsday SF stories you could make the weapons cobalt bombs. That wouldn't work for scenarios that have Earth uninhabitable thousands of years into the future. However it would work if the author only needs Earth to be uninhabitable for decades into the future. If some people were able to survive somehow for as long as a century, underground, or in domed areas cleared of cobalt 60, or because of patchy distribution of the cobalt 60, then after that they'd be able to survive fine out of doors, though with increased cancer risk for a few more decades. That would fit many post apocalyptic science fiction stories. But it ain't going to happen in our world.

DIFFERENCES BETWEEN MODERN BOMBS AND HIROSHIMA / NAGASAKI

Hiroshima and Nagasaki bombs were much lower yield than modern bombs. However that doesn’t change the conclusion, as the effects are over soon for the higher yield air burst bombs as well.

Another difference is that Hiroshima and Nagasaki were air bursts, exploding before they hit the ground. A nuclear war between superpowers would involve some ground burst attacks on missile silos and similar, designed to penetrate the ground. Those do o create longer lived hot spots, but they are also more localized. Parts could remain uninhabitable for years later like Bikini Atoll in the Marshall Islands. The inhabitants returned for a while but they found that the radioactivity was accumulating in their food, with too much of it to be safe so they left it and have not yet returned again. The main difference there is that the second Bikini atol test, Baker, was a test beneath the sea. Similarly to a ground attack, it produced a lot more radioactive debris and long lived isotopes.

So, with larger hot spots like that, near places that had ground attacks, it’s not strictly speaking uninhabitable altogether, but the risks for human health are high enough so that one wouldn’t want to live there when you can live in places that don’t expose you to those health risks.

The other way nuclear weapons could make the world a difficult place to live is by causing a nuclear winter. The early 1980s models were shocking in their predictions, of a cooling by 40 to 60 C (70 - 100 F) immediately after the exchange. Scientists thought that would happen right up to the early 1990s. However, the way the smoke behaved during the Kuwait oil fires in 1991 caused the scientists to question their models and they now predict at worst a “nuclear autumn”, with significantly lower temperatures for a few years, but they no longer predict those extremely devastating tens of degrees reductions in temperature.

Based on an extract from my: Doomsday Debunked - Nibiru Is Nuts - What About Nuclear War, Asteroid Impacts, Runaway Warming,... See it for more details and links.