Companies like Iron Mountain provide this service to businesses. Even if a bank no longer offers safe deposit boxes (and you were relying on their "safe", which for small banks was an arbitrary thing), there are usually facilities in larger cities that offer the same sort of service.

Just went through this with my family (in Australia) when the local bank branch said they were shutting down the service. It was basically a locked room in their branch, there was no "safe" involved.