How will I access a specified version of Lightroom in the future?

If this is a subscription, what happens in ten or twenty years when I want to open my old computer and work on an old photo from a previous library?

Not sure what you mean by "work on". Once your subscription ends, your Lightroom Classic library module still works 100% and you can import, add metadata, search, export your edited photos, and do everything that the Library Module allows. In fact, the only things that don't work are the Develop Module and Map Module and synchronizing with the cloud after the subscription ends.

How will I be able to access the specified version of software if I'm no longer paying for it

The old version of Lightroom Classic should still be on your old computer. Why wouldn't it be?

For what happens when the subscription ends, see

https://www.lightroomqueen.com/cancel-cc-subscription-photos/