Is that base feasible ? I'll say probably yes, but your requirements are quite hard to deal with.
First you state you need a stable area to place it. That's not that much hard to find, you just need to avoid as much as possible regions of the worlds were
tectonic plate overleaps. Since there are not billions of them you just need to avoid the areas marked on such map :
It does really limit the area where you'll be able to build your base, but you'll be happy to not have to face volcanoes and earthquakes. Though there'll still be a risk, but earthquakes and volcanoes are less likely to be far away from these areas.
Then you need it to be deep inside the seas but with sunlight anyway. As stated in the comments, water stops light after some meters, thus
it'll be hard to have natural light if you want your base to be deeply under the seas.
What could probably be done would be to place your base under the seas, and use a system of mirrors, or a stuff like that, linked to the base,
providing lights from outside, inside your base. It would be a kind of lightwell running over a few hundred meters. Don't know how much light
you could bring on like that, probably not much if your well is not located right at the surface of the sea.
Then you want your base to be far from any shore. If you can bring sunlight with a well that's not a real problem, you can find quite a lot of place
in the ocean hundreds of kilometers away from inhabited islands. The main problem would be in such places your base would be located very deeply under
You also want it to be not visible from the surface ? I have a very good news for you. Modern nuclear submarines are known to be almost impossible to detect while submerged these days, they are almost noiseless, and sonar can't detect these easily (don't remember if this is due to some kind of material or just because of the said "noiselessness"). So you'll probably need your base to be build with some of these special materials used by armies and navies to build
stealth tanks/planes/ships, but that's something that exists in our current state of technology.
After that you want a protection against radiation. You already got one just by having your base under water. Water operates as an excellent shielding against any sort of radiations.According to that article :
only 2 meters of water are required for a good protection, your base would most likely have a few hundreds of meters to protect it. That's far enough. Though if you ask a protection to survive a direct hit from a nuclear weapon I think you don't need it : your base is stealth, so you'd probably only be hit by chance. Thus you can't do much about it.
Next point, you think your base needs weapons and stuff like that. Well you could have torpedoes to fight underwater, and some guns inside the base in case of a boarding, but to put it clearly, who would ever be in a position to assault such base ? Most likely elite commandos from a random armies, and military submarines/nuclear submarines.
In such case, your few civilians friends and your are just screwed. You will never be able to fit enough weapons in a single base manned by a few people to protect against an army.
Anyway should I protect that base I'll probably use an anti submarine net, somekind of big net that was used from time to time to deny the access to a port to ships and submarines.
So that any opponent would have a hard time coming close enough to fire or board the station. Probably combined with proximity detection mines. But all of this is quite expensive.
And dangerous by the way.
You now need a source of food, water and oxygen ? Well for food you could still make things grow inside your base, such as vegetables with artificial (or natural) lighting. You could also farm seafood.
For water you could just use the sea water, for that you just need to treat it to remove its salt and any other unpleasant things. Most likely by boiling the water of and then cooling down that steam to get water without salt. Such facilities already exists, and works with solar energy. Once more that's something that can be done in 2016.
As for the air I don't know exactly the process, but nuclear submarines can stay months long underwater by recycling their oxygen, and space stations/shuttles can do the same, so you just need the same system and a massive quantity of energy.
To heat your base, two options, you count on electricity and your main energy reserve, or you could perhaps your geothermal energy ? For this, being not too far away from a volcano could be a nice idea, you'd have a very good, reusable source of heating, the same one already used to supply some server farms in northern countries (Sweden if I remember correctly).
It would also solve your problem of electricity, but if I may, using a nuclear plant would help too, generates more power for a long time. Better yet if you can find a fusion reactor then you could have an almost infinite source of energy, by using hydrogen, that can be found by breaking down water into O² and hydrogen.
Last point, you need fun, video games and stuff like that ? There are big underwater cables running through the oceans and providing internet. You could just do a little junction to one of these with your base, and with a little of hack get free internet without anyone being able to detect where you are.
I hope my answer will help you !