Is it possible to send all nuclear waste on Earth to the Sun?
Sending nuclear waste to the sun is of course physically possible, yet there is one major obstacle: energy, and thus money.
Let's consider the launch of a barrel of nuclear waste to the sun. You don't want the waste to start orbiting the sun - eventually falling back to Earth - so you must send it straight to the sun. However, Earth is travelling around the sun at around $30$ km/s so you would have to give the barrel an initial speed of at least around 30 km/s for it to stand still in the heliocentric frame of reference - the effects of the rotation of the Earth are negligible. This is two times the maximum speed of an Ariane 5 rocket.
Now, say you want to send a ton of waste to the sun. For a four stage rocket to reach this speed, with this payload, using the best known fuel - that is liquid hydrogen and liquid oxygen -, it needs to weigh around $44\times 10^3$ tons: this is more than 10 times the mass of Saturn V. Now, let's assume that your rocket's mass is more realistic, say $3,000$ tons. Then, the payload that finally reach the Sun would weigh around 100kg, and it would cost around 4 M\$ per kilogram. In comparison, based on the Yucca Mountain nuclear waste repository, it seems that storing nuclear waste underground costs around 1000\$/kg.
Finally, as you said, the rocket could be highly damaged by solar winds, so you would have to protect the nuclear waste in a steel canister. Then, only half of the payload would be nuclear waste.
It is possible, but it will be at least double waste of resources.
- The launch will be, as James answered, very expensive.
- Who says that nuclear waste will be waste forever? There is effort to recycle nuclear waste. When succesful, the waste will become a resource.
While others have emphasized the prohibitive cost (in the order of magnitude of the US GDP) and risk1, time and handling will be a problem as well. The two are related: The one common key factor in safe handling of high activity waste is, unsurprisingly, effective shielding, inevitably involving a lot of shielding mass. I'll show first that it will need an unrealistic number of launches for even almost unshielded fuel. I'll then briefly touch the difficulties involved in handling this during launches.
Mass
Let's examine the mass we need to launch. This is somewhat tricky: Are we measuring the net amount of nuclear waste? Spent fuel rods? Vitrified waste? Waste in concrete? What would change if we were sure we'd shoot it into space? Would it still need to be sealed that well? After all, the highly radioactive and highly poisonous spent fuel must be carefully separated from the biosphere also during launch. (I actually assume that shooting it into the sun is your idea of the ultimate separation.)
One number which floats around is 70,000+ tons of spent fuel in the U.S, or something like 200,000 to 300,000 tons world wide. This aligns with the number of maybe 25 tons of spent fuel per year and reactor, assuming that the roughly 400 reactors world-wide have run 30 years: 12,000 reactor-years with 25 tons/reactor/year. But this is a net weight. In order to transport and handle the fuel it is typically melted with glass or sealed in concrete, tripling or quadrupling the mass. In order to safely transport spent fuel, it is then sealed in a steel container which weighs 100 tons per 10 tons of spent fuel.2
But let's assume that we do not try to launch ordinary spent fuel steel casks. They would not survive a launch failure anyway3. Instead we just launch unshielded glass pellets or such, and hope that our rocket can handle the substantial radiation for the few hours between loading and launch. This still implies the need to launch an overall gross mass of about a million tons in order to dispose of 200,000 or 300,000 tons of net nuclear waste.
An Ariane 5 rocket can transport 10 metric tons per launch to a geostationary orbit; it will likely be less if we want to leave earth's Hill Sphere which I would consider good enough for our purpose, given a little additional push.
This results in about 100,000 launches (costing perhaps 2*10^13 dollars4, at a cost of 200 million dollars per launch). SpaceX will launch cheaper, but on the other hand higher safety requirements may make launches more expensive again.
But my main argument is that even if you launch 2 or 3 rockets every bloody day, you need 30,000 days, or 75+ years. That is, if you stop using nuclear power now. The 400+ nuclear plants produce around 10,000 tons of net/30,000 tons of gross waste every year needing 3,000 launches, which makes sending the waste to space worse than painting the Eiffel Tower: While you launch the amount of nuclear waste actually grows. You need to launch 10 Arianes every day to just keep the amount of nuclear fuel from growing.
Activity
Spent nuclear fuel is extremely radioactive. Standing next to it is lethal within minutes, without ingesting anything. (Remember the Chernobyl cleanup crews taking turns running in and out of the reactor for just a brief cleanup action?) I could not find information about the short-term impact of strong gamma and neutron radiation on electronics besides general statements that it does affect them.
Even if the launch vehicle can handle the radiation, people most certainly can't. Already for purely technical reasons it's not immediately clear to me how, where and when the payload container is assembled. It will be necessary to do that on the general launch site, possibly next to the pad, because after unpacking the fuel from the transport casks humans generally cannot come near it.
All handling after unpacking must be remote or robotic. Handling the raw pellets is something normally done in special nuclear facilities in order to avoid nuclear contamination of the environment. They are heavily regulated and audited, far beyond any civilian space enterprise. Spent nuclear fuel handling completely changes the character of the site.
The non-technical aspects are at least as challenging. Handling spent nuclear fuel is a security risk. Processing sites are high-security operations for fear of direct terrorist attacks as well as theft of nuclear inventory, either for a dirty bomb or for the Plutonium in it. There will probably be a requirement to have airplane-crash safe buildings and/or anti-aircraft defenses, at least smoke screen installations, because obviously the launch system is exposed and vulnerable. There will be regular high-volume nuclear waste transports from all over the world to the site which need to be protected as well and must overcome potential local political resistance along the train lines or highways. In order to launch 100t of vitrified waste you need to transport perhaps 10 caskets of 100t gross weight each, which roughly corresponds to a single transport train, each day. For comparison, such transports happened roughly every other year in Germany for the past 20 years. The last such transport in 2011 needed 5 days to cross central Europe and was protected by a police force of 30,000 (!). You would have that each and every day.
Heat
Some of the radioactivity is creating heat in the order of 100kW/t heavy metal for fresh waste, decreasing exponentially to 1 kW/t after 10 years. After 5 years the activity may be around 5kW/t, making the 10t payload a heat source of perhaps 25 kW (about 12 household space heaters), assuming 5t of actual spent fuel. At this point the fuel will not melt even in the absence of active cooling, but such a literally hot payload is certainly unusual for a launch system. It may affect electronics and fuel tanks. I suppose that the heat alone makes it necessary to minimize the time between loading and launching.
Conclusion
First of all the need to let spent fuel cool before anything can be done with it results in an unavoidable multi-year stockpile on earth, even if the prospect of shooting it into space sounds seductive at first.
Transport and handling spent fuel is a dangerous affair needing elaborate technical equipment and technical as well as military-grade safety measures. The payload's physical properties are unusual enough to present new engineering challenges.5
The amount of existing spent fuel stock as well as the production rate would make it necessary to launch dozens of payloads a day in order to substantially reduce the amount of spent fuel on earth at all. It would be a matter of many decades to ship the existing nuclear fuel to space which renders it not an elegant short- or midterm solution but a large-scale, long-term, excessively expensive, excessively risky military-industrial operation.
1 Even though you explicitly excluded them from your question.
2 It is worth mentioning that these containers can handle spent fuel only after it has been in a spent fuel pool for a couple of years, when they have lost some of their initial radioactivity. In other words, you cannot transport fresh spent fuel very well, and several years' worth of spent fuel will be on earth no matter what for purely technical transport reasons.
3 But they do survive a train crash.
4 The US GDP Is about 1.9*10^13 dollars.
5 This is actually the case no matter what you do with the fuel. Even simply burying it creates unusual engineering challenges because the heat and the radiation accelerate the corrosion and general deterioration of the containments.