There appears to be confusion here in definitions, and I feel you may actually be trying to solve an issue that isnt present for you. Let me explain.
You are asking for something whereby it cancels after x time automatically. Automatic cancellation or expiry is not a feature of the platform at the present time.
In terms of what you are saying there, I understand the analogy, however the analogy is incorrect. You are confusing contracts (obligation to pay) with automation (It automatically renews).
There is no obligation to pay from the point of view of the way the software is built. A customer can log in and cancel their subscription at any point, should you give the ability to do so. What it will do
Automatically generate an invoice for the next period
Automatically take payment, if you have the system set up in a manner to do so
If you don't have the system set up to store payment details, then of course payment would not automatically be taken. So if you set up, for example, stripe card payments and dont select to allow customers to save payment methods, this is what would happen.
End of the period approaches for a customers subscription
Invoice is generated by the system
Customer chooses not to pay that invoice
Invoice simply expires, and the subscription also then expires
In your example on rental, the same is also true. If you come to the end of your 5 year rental agreement, in most cases there would either be a new agreement put in place, or if unpaid it would simply come to an end. There does tend to be a notice period added to peoples agreements in these cases, and in terms of a rental there is also still a contract in a lot of countries if you are still living there and you dont leave (the contract would be implied by action), but thats a different story. The point is, a subscription and a contract are not the same thing in any way. A subscription is simply paying for something in advance of a service you receive, which may or may not have a contractual period. These do however tend to be ongoing