Yes, you can continue to use it. Most of the Professional features are not affected. Only the Professional features that require access to our servers will be disabled.
Besides the obvious access to updates, the other thing that can be affected is using remote storage services which rely on OAuth2 for authentication (out of the top of my head: Google Drive, OneDrive, Dropbox). These storage services need to refresh the authentication tokens stored on your site as often as once per hour. Due to the way OAuth2 works this requires a centralized service implemented by the software vendor. In the case of Akeeba Backup this is implemented on our own servers and we limit its use to clients with an active subscription.
If you do not plan on using any of these storage services you can still use Akeeba Backup Professional past your subscription expiration. For example, Amazon S3 does not use OAuth2, it works just fine without an active subscription. The same goes for local-only features e.g. including additional folders and databases, regular expression filtering, the Site Transfer Wizard and so on.
Do note that our software cannot retroactively support changes in Joomla, PHP, browsers or even the third party storage services which did not exist when the version you are using was published! Between all of the aforementioned change factors there is a breaking change that needs to be addressed by an update every 4 to 6 months. If you're lucky you can postpone the update for a year or so but eventually you will need to install an update.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!