Hospitals are not storing the data on a harddrive in their basement so clearly this is a solvable problem. Here's a list of AWS services which can be used to store HIPAA data:
The biglaw firms I’m familiar with still store matter data exclusively on-prem. There’s a significant chunk of floor space in my office tower dedicated to running a law firm server farm for a satellite office.
I worked as a lawyer for six years; there are extremely strict ethical and legal restrictions around sharing privileged information.