I did ask my supervisor about this, and apparently it’s because digital records could be lost or deleted. How much would it take to make them durable enough?

Assuming that there are some legal requirements that the records be stored securely, you could back them up digitally with Tarsnap for $0.30 per GB transfer, and $0.30 per GB per month of storage. If you loosen your security requirements and go with Amazon EBS, you can do it for exactly a third of that. I expect competitors to offer similar pricing, but those are the ones that I'm familiar with. Both of these are fairly low-level options, and would require someone who knows what they're doing to set up. Of course, printing everything out and storing it also takes a lot of effort, but it's an ongoing effort that everyone takes for granted, whereas getting electronic backup set up would be something new, and therefore painful.