Sort of. You don't _have_ to enable storage etc in your app. Being a 'node' can be independent of running an app. And your app can just store things on the network without having to worry about making a user share their space (and do onboarding to the network etc).
> If so, what happens to other people's data when I turn off my machine? What happens to my data when other people turn off their machines, or are otherwise unavailable?
If you turn off your storage, you stop earning money. The network ensures there are X (currently 8, I believe) copies of each piece of data available at any one time, so data on the network would be replicated to make up for whatever went missing as you (or anyone) turn off a node.
> I don't understand what problem this is solving
The problem of centralisation and data hoarding. Facebook. Google. Being de-facto forced into giving up your data as a user.
It aims to decentralise this, and (hopefully) will make it easier for users to remain in control of their data, without the need for massive server farms etc. By using spare space on user's devices.
The crypto currency is an incentivisation layer. (you are rewarded for participation in storage / serving data etc).
I'd recommend reading the primer (https://primer.safenetwork.org/) to get into the details of how this will work.