There are many different ways, depending on the infrastructure of the application. Network Attached Storage (NAS) solutions from companies like NetApp are common. For other applications, enterprise SAN arrays may be appropriate using iSCSI or FibreChannel.
There are even more creative folks on the hardware side, such as that detailed in this blog post from Backblaze about their Storage Pods, 67TB of NAS for less than $8k (probably even less now with falling drive prices).
Other considerations include redundancy and backup. Performance considerations may affect whether one uses enterprise arrays with SAS drives, or commodity SATA hardware.
Obviously any additional hardware comes with considerations for power, space, and connectivity. These may be handled by the company if they run their own datacenter, or they may be handled by another company via colocation or a fully managed environment.