You can set up remote access to a NAS.
However there are a few caveats,
Generally speaking, a NAS works reasonably well for streaming and file sharing in a home network environment. Problems occur when the network connections are too slow or the NAS processor can not keep up with the demand for large file transfers.
The ideal way to have a NAS connected to data intensive devices is via Gigabit Ethernet.
Although WiFi is convenient for mobility, wireless interference can cause it's connection speed and quality to vary wildly.
Wireless signals are affected by many factors including wall density, electrical interference, directional antenna range, etc. As wireless connection quality varies the connection speeds are renegotiated. The connection may start at 150 Mbps and end up at less than 10 Mbps or drop entirely.
For either Ethernet or WiFi, the connection speed will always negotiate to the speed of the slowest device.
For example, a super fast 300 Mbps wireless n router can only connect to a wireless g laptop at a maximum of 54 Mbps.
In order to achieve Gigabit LAN speeds, all devices in the connection need to be Gigabit. This would include the LAN ports on the router, the network interface cards in both devices and the cables connecting them.
That being said,
If you want to access your NAS via the Internet, you will need to set up remote access via port forwarding on your router.
The main problem here is, all data that you want to access needs to be uploaded via your ISP connection. Upload speeds are usually lower than download speeds. Most broadband Internet services including cable, DSL and fiber are asymmetric. which just means that more of the bandwidth is dedicated to the download channel than the upload channel. For instance 6 Mbps download and 1 Mbps upload. This is due to the fact that most users download much more than they upload.
Another factor is your WAN IP address.
In order to access your device remotely you will need the WAN IP address that your ISP has assigned to your connection. You can see your current public WAN IP address here: http://www.mywanip.com/
Most ISPs use periodically changing (dynamic DHCP) WAN IPs. So you will need to get a static one or use a DDNS service. http://dyn.com/dns/
This will allow you to always be able to access your device via the Internet even when your public WAN IP changes.
I personally use three different brands of dedicated NAS devices, two are configured in a raid one configuration for redundancy, you can even have the device email you when a drive is failing, and swap in a replacement without losing a bit of data.
If you're in the market for a NAS, I would highly recommend the Synology line. I use the DS212j http://www.synology.com/products/product.php?product_name=DS212j&lang=us By far the best user interface and features. It runs Linux and is highly configurable. It includes a download server, media server, individual user security, and too many other features to list. The Synology DS212j is now under $200 at Amazon. http://www.amazon.com/gp/product/B005YW7OLM/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=B005YW7OLM&linkCode=as2&tag=excharge-20
Good Luck...