Is this a consequence of decimal counting? Is the hypotenuse always irrational for any counting base (binary, etc, etc)?

I suppose you could count things in square roots of 2. The hypotenuse would be 1 unit while the remaining sides would be 1/Sqrt[2] units. :p

Does that have anything to do with the invention of the radian?

Yes, but I think we are confused enough.

Imagine I draw a line on a piece of paper, and I measure it.

First I use a ruler calibrated in cm, the line between 4 and 5 cm. Then I get a better ruler, marked in mm. Now I know that the line is between 4.2 and 4.3 cm.

OK, so I get a vernier gauge and measure it again. The line is between 4.25 and 4.26 cm.

I can carry on doing this for ever, using finer and finer scales but, I never get to the point where the line stops exactly on one of the marks of the ruler. If it's too close to say if it's more or less than 4.255 then I just need a finer ruler, one that tells me that it's between 4.25500000000000003 and 4.25000...4.

Almost any line I draw has this property of being irrational.

The weird lines are ones that are exactly 1.603 units long or whatever.