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Would the people running nodes need bigger servers?

Definitively yes.

In fact, the raw data of storing the blocks doesn't even matter that much.
It's more about organizing that data into indexes and distributing it via API to others.
This costs a lot of RAM, and RAM is expensive.

Also the more people are connecting to Hive nodes VIA the API the more bandwidth gets used.
Internet bandwidth is also expensive and adds up quick.