Why does the world store nuclear waste and not
just shoot it into the Sun or deep space?

One is that this is very, very expensive. When the Parker Solar Probe was sent to take measurements of the Sun this year, it cost US$1.5 billion just for a spacecraft the size of a small car to get that far.

It seems simple to shoot an object towards something as big as the Sun - which is 1.3 million times the size of Earth. But it’s actually very hard. The Parker Solar Probe (a NASA robotic spacecraft en route to investigate the outer corona of the Sun) has to swing past the planet Venus seven times to slow itself down enough to get close to the Sun.

The other reason is that rockets sometimes blow up on the launch pad, or in the atmosphere. This would release the waste into the environment and make the problem even worse.