There are a bunch of things to talk about here.
First thing's first. Size. You need to take a look at how much data you have and how much its grown. Based on that, you need to estimate how much it will grow in the future. You need to know this information, as it will determine the hardware you need now and in the future. Without getting a good sense of this information, you might go with one solution and find it cannot sustain your data growth. Its always better to overestimate, than to underestimate. There is no point in buying a hardware NAS that supports only two 4GB drives, when you expect your data to reach more than 8GB.
Next is RAID. All RAID configurations, except RAID-0, provide redundancy. People often misunderstand this concept. Without getting technical, redundancy provides continued availability to data. This means, if a drive dies you can still read and write your data. Redundancy does not protect your data. If more than one drive dies, data gets overwritten, data gets corrupted, your data is gone. RAID is not a replacement for backups. If the data is important to you, you will make proper backups.
As for RAID controllers, you dont need to worry about that. NAS hardware will either come with either come with hardware RAID controllers built in, or the RAID will be done created by the server software.
There are a variety of NAS appliances out there. Higher end "prosumer" devices like Drobo work, but are often quite expensive. The more storage they can hold, often means the price can grow quite high. These higher end NAS appliances often use proprietary RAID technology that allows you to add disks to them after the array is created, which you cant do on standard RAID arrays. Obviously this has advantages, but locks you into their ecosystem. Hardware reliability is another issue. I dont know about Drobo's current hardware, but in the past there have been many complaints about them. The higher end devices often come with lots of additional features, which you may or may not use.
Lower end NAS appliances do exist, some of which do RAID, some are JBOD. Again, you need to look into their cost, reliability, and size limitations to see if they are right for you.
The real advantage of a hardware NAS appliance is size. They are incredibly compact. You also get support, but once the warranty expires, your mileage will vary.
I am a fan of building your own NAS. You can turn just about any PC into a NAS with little effort. Operating systems like FreeNAS are extremely robust and almost limitless in scalability. You can add internal or external storage to increase its pool. Some people find FreeNAS to be overkill. There are a variaty of free NAS operating systems. OpenMediaVault is one I highly recommend. Its lightweight and easy to set up.
I had an old PC that had room for a surprisingly large number of drives. I packed it full of 8 drives I had laying around and installed OpenMediaVault onto a flash drive which that PC boots from. This NAS system literally cost me nothing. While much larger than a Drobo, you cant beat the price.
As for remote access, thats easy. Many of the "prosumer" appliances give you access to your data through their websites. They dont copy the data, the appliance just connects to the website, therefore you dont have to set up any services on your own.
All the NAS software I have seen offer a variety of methods of access, such as web, FTP, ans streaming. Using dynamic DNS is perfect for these setups. Even ISP provided routers often have DDNS built into them now for home users.