Visualisation without a chess set

Not necessarily. The more you do it, the easier it will become. It's not something you can nail in 1 day. As a kid we used to do thousands of tactics from books with diagrams. I think visualization started there for me. Of course there could be a period of time where you might have to adjust for having real pieces in front of you if you've only ever done chess online and vice versa. 

Also, as with everything, some people might find it easier to learn than others. I know plenty expert level players that can play pretty well, but do not have the visualization skills needed to play a blindfold game.

P.s.: oh btw, there are tools online that could help you train visualization. One of which is also on this website.

I have done 4000 puzzles on Tactics Trainer, but despite this when I close my eyes I am unable to even picture a square, let alone a chess board. Am I lacking even something more basic than board vision?

There are some visualization tools out there (and one right here), but I wouldn't say they are that good. Overall, I think it's one of the issues right now. Visualization training is not nearly as developed as it should be.

Not necessarily. And you may not be lacking anything, much less so board vision. Have you ever tried to follow a game without a board? It helps, even if it can be frustrating at times.

When you say 'follow a game without a board' do you mean for example configuring a chess-playing app to give verbal moves? 

No, i mean... say that I give you this PGN

1. Nf3 Nf6 2. c4 c5 3. g3 g6 4. Bg2 Bg7 5. O-O O-O 6. d4 cxd4 7. Nxd4 Nc6 8. Nxc6 dxc6 9. Qxd8 Rxd8 10. Nc3 Be6 11. c5 Nd7

Can you visualize the positions? Can you follow the game without looking at a board, without moving pieces? You may want to try...

I am afraid I can't. But you have given me an idea. Some non-interactive chess books give a diagram after a few moves. I could try reading a chess book and covering up the diagrams. 

That's a good idea. That's how I started with chess visualization.

I wouldn't say visualisation is necessarily about picturing the board and pieces in your mind. If you can simply understand a position in a few moves time then that is also visualisation. If you can look at a position and think, 'if I move the knight to d6 and the bishop to e8 then i can pin his rook and block his queen from the diagonal' then you are visualising. I can't see a board if I close my eyes, but I can imagine how a position will change if I'm looking at a board.

That's also true. OP talked about the ability to close his eyes and visualize the board, which is only a way to visualize.

The language is vague, so it's hard to know what's going on in other people's heads.

I've played plenty of blindfold games... but I'm not very good at it. I can't see a whole board, I only see the area or few pieces I'm concentrating on at that time. Or for example I'll see the pawn structure but not the pieces.

I agree with the poster who said this probably happened for me after solving a lot of puzzles (hours a day, every day).

In fact during some tournaments the 3d board was so distracting to me that whenever I needed to calculate, I'd look down at the edge of the board and visualize the sequence as a 2D diagram.

This is a good way to do it.

And it's easier to start with endgames because there are only a few pieces. Visualize using notation as far as you can, then when you start to lose track play the moves on the board so you can see what the correct position is.

You may have something there. I have read that studying endgames helps calculation skills. But maybe it is also possible that folk who enjoy endgames enjoy them because they are good ay calculating. What came first? The chicken or the egg? 

I enjoyed endgames, and at the time I described myself as "lazy" -- I really didn't like calculation.

After going through an endgame book (and working on tactics) my calculation improved though... this was almost 10 years ago

     Visualization is one of the most important "talents" for chess players. Unfortunately, it is given out unequally at birth. Certainly practice will help maximize you ability, but it is doubtful that the kind of board you usually play on will make much difference. So you will probably be stuck--like and most other players--at the sub IM, GM level.

i just did this knight exercise but i still can't play blindfold chess PLZZZ helllp

oh yes i know that one well

Keeping this image in my head for when I can't sleep at night.