The typical and most reliable procedure for managing radioactive waste is vitrification: creating a purified mixture of molten glass and then introducing an evenly distributed non-critical ratio of hot waste material into the glass, and allowing it to harden into a solid glass object. The radioactive glass object is then carted off to a repository, for permanent storage, in accordance with the half life of the waste, which might be centuries or more. Vitrification is a safe way to prevent accidental criticality, so that all the waste stays cool and is easier to shield.
Generally underground storage sites are the most desirable locations for the final resting place of vitrified waste. This provides a simple barrier to the penetrating radiation that the waste may emit.
Security is essential to the storage of radioactive waste, since unaccounted waste means there's some nasty stuff floating around. This adds effort to maintaining a site.
Ventilation is necessary, since ionizing radiation produces an accumulation of fee oxygen and hydrogen by catalyzing moisture in the air. This means offgassing equipment is needed to ventilate the natural accumulation to prevent explosion hazards. This adds complexity to storage.
Degradation of construction is a long term pest, in that the site must be constructed of high quality, durable architectual members, equipped to last centuries, and not collapse within decades. This adds expertise and expense requirements.
Site selection should be a no brainer though. Consider that Ukraine can make some decent income off the tragedy of Chernobyl, given that they have an unusable sector of their territory relegated to the reactor sarcophagus. Yeah, the sarcophagus is impossible to manage above ground, but what about digging underneath it and excavating a massive permanent waste repository, and charging money for depositing waste there? Nobody wants anything to do with Chernobyl. It's a ghost town. Seems like a chance to employ the site as a massive underground waste repository.
Same goes for Fukushima. Take a geological survey of the site, design durable, earthquake-proof architecture for an underground repository, and charge money to dumpwaste there.
After construction completes, your budget mostly comes from staffing qualified nuclear engineers and security personnel. Little else is necessary. A nuclear reactor and research lab can provide power to the site and provide an intellectual basis to attract new staff. Doesn't this sound like a sustainable plan?
In America there has been this massive battle over Yuka Mountain. It's politically hazardous to store waste underneath otherwise uncontaminated land. The protest generally stems from the not-in-my-backyard philosophy. There are tons of superfund sites, that are doomed to contamination for decades because of simple bureacratic laziness. Most of them are pretty close to cities. I think America could probably find sites, but they usually get locked up in legal messiness that blows any deal. I think there are probably places that could accept waste, and there's no rational reason to care, but people fight it anyway, because everyone seems to enjoy irrational litigation as political sport as a sort of clerical version of new-deal make-work contruction projects. But I digress.
There are reasonable ways to confront the challenges of radioactive waste storage. Obstinate people use this objection as an example of an insurmountable challenge simply because they're stubborn.