Is it feasible to store spent nuclear fuel on an uninhabited planet, moon, or in space in the near future?

Technically, it already was and is feasible but the problem lies in the economic aspect. Simply put, radioactive waste can be disposed of quite safely by launching it into Earth's orbit and then "shooting" it out of the Earth's gravitational field. This requires a rocket and a simple device to launch a capsule containing radioactive fuel. If the capsule has a higher velocity than the second cosmic velocity or approximately 4 km/s in geostationary orbit, it will leave the Earth's gravitational field and fly into space. This seems like a large number, but considering the lower gravity and absence of drag forces, it's not a technically challenging task — at least from the perspective of a physics teacher :D. It might be possible to aim the capsule so that it lands at a selected location (the moon, a planet, the sun).

So why aren't we doing that now and instead dealing with deep geological repositories? Simply because launching 1 kg (of anything) into Earth's orbit is expensive. It's estimated that the cost of that kilogram is over US,000. Although costs are decreasing with the advent of new rockets (Saturn rockets cost US,000/kg, new Falcon Heavy rockets cost around 2,000 USD/kg in 2025 and Starship rocket predictions may even be below US,000/kg), it is still a very financially demanding method.

The cost of building a deep geological repository ranges from USD 500/kg (Czech Republic estimate) to USD 2,000/kg (Switzerland) and USD 2,500/kg (France). At current prices, it is therefore at least 4 times more advantageous to manage spent fuel thru deep geological repositories. This method also has the advantage that we have control over the radioactive material throughout the entire process. While the transportation of materials into space is well-managed, it is not perfectly reliable. There is a risk of an accident, which could lead to an uncontrolled release of radiation. That's why flights of rockets with radioactive propulsion are very limited.

Another reason for leaving spent fuel on Earth is the possibility of its further use in new-generation reactors, thereby extending the lifespan of nuclear power plants.

Just for interest's sake, I'll add that in the event of significant progress in alternative methods of transporting cargo into orbit (space elevator, space catapults), the cost of transporting 1 kg into space could be reduced to around USD per kilogram! Then it would be much more advantageous to "throw" radioactive waste into space.