To answer your question, NAS and SAN are usually not servers in the typical sense of the word. A NAS is typically a headless server running some variant of Linux that is basically a bunch of hard drives with an ethernet port on the back. You access it the same way you would access a share on a Windows machine. NAS is typically IDE or SATA.
A SAN, is typically SCSI, and a lot more expensive. Basically, the typical SAN setup is a 3 or 4U rackmount box with a bunch of hotswap SCSI hard drive bays. The SAN box connects to one or more servers with SCSI. This is useful if you wanted to cluster your SQL server, in the sense that you would have two servers running SQL, and both of these servers would connect to the SAN box for data storage. This way, both servers are using the same hard drives to store the DB files.
There are other more elaborate types of SANs that involve fancy fiber channel switches, but I don't have much experience with that type.
I should note that Dell made some PowerVault series NAS boxes (770 & 775N I think) that were actual real servers. Some newer NAS boxes also run Windows Storage Server. But in general, no, you can't run applications on a NAS or SAN. You need a separate server for that.
Just to clear it up, SAN stands for Storage Area Network and NAS stands for Network Attached storage. So the difference is that a NAS is part of a network (a LAN in the case of our homes) and a SAN is a network itself, that can have different stationts like the ones FredWeston described. So for your home, you will be looking for NAS.
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