Is it possible to send all nuclear waste on Earth to the Sun? If we neglect the danger of unsuccessful lift-off of the rocket and the cost, would it be physically possible to send all nuclear waste on Earth to the Sun?
Will there be an obstacle that prevents this? For example, solar winds?
 A: I answered a similar question elsewhere on the SE network. This is a slight update to that $2\, \frac 3 4$ year old answer.

Will there be an obstacle that prevents this?

At a bare minimum, there are at least five obstacles:


*

*If you wanted to do this, sending it out of the solar system makes more sense.

*We can't even do that. We can't even make it crash into the Moon.

*Even if we could to that from a technical perspective, it would cost too much.

*Even ignoring the technical and financial obstacles,it would be too risky.

*Finally, we just don't want to do this.



Sending it out of the solar system makes more sense.
Sending something into the Sun is extremely expensive; the delta V cost is extremely prohibitive. It would cost significantly less to send something outside the solar system than it would to send something of the same mass into the Sun. It would cost even less to make that something crash onto the  surface of the Moon. I'll use that as the metric as opposed to sending our waste into the Sun.
We can't do even make it crash onto the surface of the Moon.
The nuclear industry generates 2000 to 2300 metric tons of waste in the form of spend fuel per year (Source: Nuclear Energy Institute).
I'll take the Falcon Heavy as a baseline. It will be able to put 13,200 pounds of payload into a translunar injection orbit. At a minimum, that means 384 Falcon Heavy launches per year. Note very well: That bare minimum means stacking the waste, unprotected, on top of the Falcon Heavy launch vehicle. That is unrealistic for a number of reasons. Being completely unrealistic and assuming that there's only one pound of encapsulating material per pound of waste, that means over two launches per day, every day of the year, just to keep up with the current production rate of nuclear waste.
To make matters worse, that 2000-2300 metric tons per year is the tip of the iceberg. Per that same source, we have accumulated almost 70,000 metric tons of used fuel. There's also contaminated water, contaminated housing equipment, contaminated control rods; etc. There's a whole lot of stockpiled nuclear waste. There's no way to get rid of it in space.
Even if would could do it technically, it would cost too much.
Using my ridiculously optimistic factor of one pound of encapsulating material per pound of waste, getting rid of that 70,000 metric tons of used fuel would mean a cost of almost 2 trillion dollars. That number is ridiculously optimistic. Plug in any realistic number and the resulting cost will make the war on terror look like very small financial potatoes.
Even ignoring the technical and financial challenges, it would be too risky.
My factor of factor of one pound of encapsulating material per pound of waste is ridiculously optimistic. Look what NASA has to do for the radioisotope thermoelectric generator it uses for deep space probes. Those RTGs are built like tanks, to the nth degree. Rather than my one to one factor, a more realistic factor would be tens to one, or more. That would make the wars on terror, on poverty, on cancer, on whatever look like microscopically small financial potatoes.
We just don't want to do it.
The ability to have multiple beyond Earth orbit launches per day is every space fanatic's dream. Wasting that fantastic capability on launching nuclear waste into space? That's every space fanatic's nightmare.
A: 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.
A: Yes it is. But consider the sheer amount of radioactive waste, e.g. Radioactive Waste Amount

Short answer:  Stored waste in the U.S. is roughly 60,000 tons
  (high-level waste).  Take that and multiply it by four since the U.S.
  has 104 of the 400+ world nuclear plants and we get 240,000 tons.
  I disagree you would have to send stuff directly to the sun. Once it is out of Earth orbit, it's gone anyway. So let's assume basic launch costs (although they are only to orbit, you need to get out of orbit, so it'll be more expensive). e.g. Launch costs
  $27,000 per pound

That is just WAY too expensive for a waste disposal. I would rather consider drilling a VERY deep hole, say 10km, if you want to just forget about the waste forever, it'd be way cheaper for sure.
A: 
Will there be an obstacle that prevents this? For example, solar winds?

The solar wind is effectively harmless to most spacecraft unless they are exposed for very long periods of time, in which case the conductive glass over the solar arrays are usually the parts sustaining the most damage (i.e., the glass slowly turns "black" which reduces the efficiency of the solar arrays).  The bigger issue would be ablation from high energy photons near the sun.  If not done correctly, the body of the spacecraft would completely vaporize while still well above the photospheric surface (i.e., the solar surface).  That is, if your intent was to trap the radioactive waste in the solar gravity well.
The Solar Probe Plus mission is facing similar issues, i.e., how to get close without ablating.  You might find the links on the mission webpage useful.
Side Note:  If the spacecraft ablated before getting trapped by the gravity well, the effects to Earth would not even be measureable.
If you just adiabatically expand one ton of pure uranium over 1 AU in a spherically symmetric way, the density will change as $\rho \propto r^{-2}$.  In solid form, uranium has a density of ~19.1 g/cm3 or ~19100 kg/m3, so one metric ton (i.e., 1000 kg) of uranium would initially occupy a volume of ~0.0524 m3 or ~52.5 liters.  If we now decrease that density by the $r^{-2}$ factor (assuming a conservation of mass) over 1 AU, then at Earth the number of uranium particles per cubic meter would be ~10-24 m-3.  For comparison, the typical proton number density in the near-Earth environment is ~10 cm-3 or ~106 m-3.
In summary, even if the craft completely ablated at low altitude near the sun and was picked up by the solar wind, by the time it reached Earth there would be no measureable effects (i.e., we could never measure number densities that low).

If we neglect the danger of unsuccessful lift-off of the rocket and the cost, would it be physically possible to send all nuclear waste on Earth to the Sun?

I agree with @Luaan's comment in that we are already finding use for the so called "spent" nuclear fuel rods, e.g., look up articles on breeder reactors.  With the appropriate shielding, they could also be used as long-duration low heat sources reducing energy costs etc.  Of course, the concerns about inappropriate uses (e.g., for dirty bombs) are always present but there are ways to make extraction exceedingly futile.
For instance, the current containers used to store spent fuel rods can withstand forces produced by a supersonic impact with solid concrete, burning napalm exposure for upwards of 24 hours, most high explosive ordinances, etc. without losing their containment capacities.  Basically, the amount of force/energy required to breach one of these containers would concern me much more than anything within the containers.
A: Tl;dr: There is about a million tons of spent nuclear fuel on Earth (in a form that can be handled). Shooting all of it into space is a gigantic, unprecedented endeavor facing prohibitive technical, logistical and security challenges. It will take decades. Its cost is comparable to the GDP of the U.S. In any case, as long as there are nuclear power plants a revolving stockpile of many  thousand tons of fresh nuclear waste must remain on Earth while the waste is cooling down.

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.
A: 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.

A: Once technically feasible and safe, which even if possibly not today it will doubtless be one day, why not just drill a mile or two roughly down the plane of a destructive plate margin at the bottom of an oceanic trench and drop thousands of tons of high-level waste down a steel or titanium pipe before sealing it with half a mile of concrete or similar? Then let the Earth automatically convey it into the Mantle over tens of thousands of years.
A: In addition to what other people have said it seems much cheaper to use a railgun for this purpose. Railgun is an electomagnetic launcher. If it would be pointed straight up and located near equator you could launch payloads to the sun during sunset everyday.
