I'm making my Son a loft bed and ladder using the same stuff and I just worked on a bar re-furb and they used LOADS of the stuff.
The fittings are called "tube clamps" but sometimes other terms.
It's a fantastic method if you want that look (I love it) and very easy and quick.
Check out here.
Your critical measurements will be the mattress size and how flush you want the frame to be to it.
You have to take into account that there is a certain amount that the tubes push into the bracket before they reach the end stop which you need to calculate.
I don't know the size of your mattress or how much the tubes push in so can't really help there.
Other than that the height of the bed and the head/foot boards are up to you.
The clamps do come in different sizes to fit different tubes gauges but the scaffold size is about 65-66mm (I think, from memory).
To calculate the amount of stock you need to order and the design of the bed you're going to need:
x2 for the full length (sides) of the bed base. Same size.
x4 for the width (ends of bed base and the tops of the foot/head board). Same size.
x2-3 of the above to go beneath the the mattress to support whatever frame you'd make. Same size as above.
x4 legs/bedposts. They would go from the floor and through the t-clamps to the top of the foot/headboard and length depends on how high you want them.
In the the example you posted they have used x5 upright (jail bars) at the top and bottom of the head/foot boards. (x5 short, x5 long) and length depends on how high you want them.
One tip I got from the guys on site was that cutting the ends of the tubes flush and square was quite important.
They used a drop saw with a metal cutting blade which are noisy and throw a shit load of sparks out (as in the pic).
I'm going to make a wooden jig, a bit like a mitre block to fit the width of the tube and clamp the tubes in with a 90 degree guide/slit to guide the hacksaw.
Not sure if it'll work but I'm giving it a try.
