If you're speaking about the stable place of storage itself, it should be on a geological timescale, so more than likely more than enough (as far as we know about geology). If you're talking about the storage facility itself, it's more variable obviously. But also keep in mind that nuclear waste storage has multiple levels of danger, not everything is depleted or enriched uranium.
Pretty much anything that comes into contact with ionising radiation is considered nuclear waste. It goes from medical equipment used around radioactive sources, to a wrench used inside a radioactive part of a nuclear power plant, to actual fuel rods.
Because radioactivity decays over time, and we know the half life of all known elements we can calculate how long they need to be stored for.
Low level radiation items and short lived intermediate level radiation (for up to 30 years) items are usually stored overground or at near surface underground caves for some time until safe to be disposed of. These represent around 90% of nuclear waste.
Long lived intermediate level radiation and high level radiation are stored deep underground (between 250 and 5000 meters) in stable and safe materials (clay, salt, etc) and are also contained in man made canisters made stuff like copper, concrete and bentonite. These are to be stored over geological times. There are obviously issues to be aware of, like how to communicate the presence of nuclear waste to future generations 10000 years in the future, or human error in handling the waste. But it's still safe enough to not be a major issue.

To reach a long term nuclear waste disposal you'd have to take some massive conscious effort. We're talking about at least 500m deep mines. Definitely not something you'll stumble upon on your Sunday walk.