Well, from what I remeber there are two main options. Firstly, like you mentioned, simply storing it underground in properly sealed caskets. Done properly, this should pose minimal risk.
Secondly, there is the possibility that if Breeder reactors come back into favor, the waste can simply be mostly taken care of by using it as fuel for the reactor. And I believe the main reason for not directly moving to renewables is that without a integrated power grid over a large area (think at least Europe size" and a very large amount of renewables, there is not enough power to consistently supply baseline power. Then again, I am not an expert, so I might be wrong.
I feel that, even if nuclear reactors are built to be much safer, the risk of nuclear never justifies its convenience (think Fukushima & Chernobil). Radioactive pollution is just too destructive. I don't see why we cannot have a decentralized grid of renewables. Something along these lines would not only be pollution free, it would also make people more energy independent and the grid more resilient. A win win situation without the risk of nuclear disaster forever hanging above our heads.
While I agree there are risks, the chance of any accident is not huge, and even if there is an accident, the chances of it being a huge accident are almost nonexistent. The thing is that fossiil fuels also have their own risks, espeically oil spills. Besides, their emmissions contribute global climate change. Point being, personally I feel that the best way to deal with increasing energy needs is to increase our nuclear capacity instead of using more fossil fuels, until such a time that fully renewable energy is feasible. I do like the the idea of a decentralized grid of renewables, but again the problem is that not all places have access to the same renewables. In a place like Finland during the winter, there is really nothing except the wind, and how do you get electricity when the wind stops blowing?
I would prefer no risk over an infinitesimal almost non-existent risk. Based on the examples I mentioned, when nuclear goes bad, it goes really bad and it affects countless future generations and radioactivity travels and is practically impossible to clean up. And innocent people end up paying with cancers and all sorts of nastiness. So why run a risk? (Not to mention that the military makes use of radioactive waste for manufacturing weapons with so called "depleted uranium" which is not considered a nuclear weapon but which causes radioactive poisoning with devastating consequences where it is used, such as in Iraq.) Not all places need to have access to the same renewables for a decentralized grid to work. Your example of Finland is quite extreme. We can start with places where there's quite a bit of sunlight, which is a lot of the populated world and then focus in more challenging places. In Finland there may only be wind in the winter but what are the chances that the wind will stop all over the country at the same time? Also, it may be possible to take advantage of tidal energy and geo-thermal. I'm well aware that moving over to renewable energy sources is challenging and expensive. But it is a price well worth paying for a resilient, non-polluting and risk free future. In fact it's the only solution if we ever want to arrive at such future.