Some hosting sites advertise “unlimited” storage, but the fine print generally excludes “abusive users” from this policy. For web hosting, they’d probably consider backups of non-website data to a service intended for basic web hosting to be abuse.

Unless they have a home lab (sounds like they don’t) or fancy expensive contract with a large cloud provider (unlikely), this is asking for trouble. Nobody offers unconditional data storage for free, its always a loss leader for another service and abuse will eventually get you banned.