There is toxic waste, but that is mostly a political problem. For starters, we are bound by treaty to not reprocess our waste, and thus we generate a lot more of it than is necessary, and a lot more of it than other countries do. The way we do it now, it is kind of like throwing away a rechargeable battery after draining it the first time. It is 100% possible to address safety concerns with current technology, but sometimes the political will isn't there to do it right (Chernobyl), or the operator cuts corners due to cost, gambling that a worst case scenario will never happen (Fukushima). If any industry deserves oppressive regulatory oversight, it is the nuclear power industry. The risks are admittedly high, but it remains the best base load solution with zero carbon emissions.
The storage isn't really that hard, just another political mess. Our one underground facility for storing waste (WIPP) had to be out of commission for a couple of years after a barrel of nuclear waste exploded and contaminated everything underground. But it worked perfectly from another point of view: There was no radiological release above ground, and the barrel exploded safely underground. If anything, this highlights the need for more capacity, but some activists have used the incident to try and shut down underground storage. This is completely backwards in my view as if that barrel had been above ground, it would have been an environmental and public health catastrophe.
Also, that particular barrel came from Los Alamos National Lab, and wasn't related to nuclear waste streams from power production, but from weapons. Someone used the wrong type of kitty litter (yes, seriously) as an add-in and it caused a chemical reaction. Edit: Point there is, even if we abandon nuclear power, unless we also abandon nuclear weapons the waste stream issue still exists and must be dealt with. And as much as I would love to be out of a job because the entire world disarmed their nukes and sang kumbaya, it simply isn't happening anytime soon.
