None if you aren't working from the drives, and only using them for media storage.
The problem with gigabit is that it's limited to about 120MB/s throughput, which is a lot slower than an internal drive.
One big issue with NAS is that many cheap GbE NICs are CPU hogs - in addition to the hard limit of ~115 MB/sec on throughput.
Newegg sells $30 Intel GbE NICs, and $120 GbE "Server" NICs (single port). The extra $90 buys processing power in the NIC that offloads the CPU from much of the TCP/IP protocol overhead. You need those "Server" NICs in both the client and the NAS box - but many of the cheaper NAS boxes have cheap NICs and can't even come close to 115 MB/s.
Of course, "uncompressed" BD movies are only about 50 Mbps, so virtually any NAS solution will be fine if ripped BD movies are what you need.
Current Infiniband cards are 40 Gbps, so forget about using them with the new Mini Pro. Fibre Channel is 8 Gbps and 16 Gbps (in common use), so they are a very expensive possibility. 10 Gbps iSCSI is another expensive alternative SAN. (These are expensive on the clients, and often very expensive on the server. I just bought a couple of 20 TB 16 Gbps arrays for about $20K each - had to spend that December quarter budget!)
Probably the biggest advantage of NAS is that it's shareable - any computer on your network has read/write access since the filesystem is owned by the NAS. SAN is not shareable without special cluster software (like the dearly departed XSAN) - SAN disks (LUNs) appear as local devices to the clients.
One disadvantage of NAS is that since the server is exporting its private filesystem using (usually) CIFS or NFS - you're limited to the capabilities of whatever filesystem the NAS box is using. I tried using a nice Linux-based NAS box for my CD collection. Unfortunately, all of my music files had Unicode filenames - and the Linux in the NAS box was smashing the filenames to 8-bit ASCII. Horrible. And, of course, there's the issue of 32-bit filesystems that won't do files larger than 2 GiB or 4 GiB or 2 TiB.
There's no right answer to the SAN vs NAS question - it depends on what you're doing. I find that iSCSI is usually the best solution for me for external "local" storage for a workstation. The iSCSI link is usually a separate 1 GbE server NIC unless the iSCSI is for backups or archiving only. In a few cases it's 10 GbE if the system has serious IO needs.
For shared storage, a NAS running the same OS as the clients is fine. At both work and home that's a Windows Server 2008 R2 x64 server (that's Win7 server) running server NICs.