From a technology perspective, yes, we can send spacecraft carrying some payload past Mars. Numerous space missions achieving this have already been completed, others are in progress, and more are planned. Several objects launched from Earth are even on track to leave/have left the Solar System, with the earliest of these ( Pioneer and Voyager missions) having begun in the 1970's.
Since material can be sent far into space, there must be reasons why it isn't. One of the primary drawbacks is cost. The current best price for sending payload to Mars is offered by Spacex at $62 million for 4 tons ($15.5 million per ton) on a Falcon 9. Per government estimates, the United States alone is expected to produce 2,200 tons of spent nuclear fuel (i.e., material from nuclear power reactors which the plants can no longer use efficiently), every year. This is equivalent to 550 fully-loaded Falcon 9 rockets, meaning that sending that waste to Mars would cost approximately 34 billion dollars, again per year. Sending it further would cost a bit more because more of the mass of each rocket would need to be rocket fuel, leaving less capacity for the nuclear waste. Perhaps $34 billion does not sound like that much, but realize that only new waste and only the USA is included in that calculation; the US also has nearly 100,000 tons (25,000 Falcon 9's, more than $1.5 trillion in rocket launches) already in storage. In addition, many other countries also use nuclear energy, many to a greater extent than the US.
This cost estimate also does not account for the potential of failures of rockets. The costs of a rocket failing on the launch pad or near Earth would, at the very least, be enormously expensive to clean up, and could be catastrophic to large swatches of the surrounding area. Radioactive material can linger in the environment for thousands of years after release, and such an accident could make a region dangerous for humans for significant amounts of time. [As recent examples, see accidents in New Mexico in 2014 and Fukushima in 2011. Chernobyl and Three Mile Island are famous examples from earlier times.]
So, with the present technology, we could send nuclear waste beyond Mars, but the costs and potential risks of doing so make such a proposal practically infeasible.(This does not mean that using nuclear power is worse than other methods of generating electricity. Depending on how one values various aspects of energy production, nuclear can be considered safer than many other methods.)