Premise 1: All straight lines have the value of length equal to the numerical value of the end point, provided the starting point of the line is assigned the numerical value zero.
Premise 2: No point can be assigned the value y.xxxxxx... or y.abcdef....(Example: 8.9999.... or 8.39465..). We can either have the point with the assigned numerical value to be either y.xxx, y.xx...x, y.abcdef..g, etc, but not y.xxxxx.... or y.abcdef....
Premise 3: All lines have starting and ending point, thus they have the value of length equal to the numerical value of the ending point. From the premise 2, the value must be either y.xxx, y.xx...x, or y.abcd...g, etc.
Premise 4: Diagonal of the square whose side is the unit of length, has got starting and ending point. Therefore, the length of the diagonal should be a value which can be expressed as the fraction with terminating decimal form.
By this argument, length of the diagonal ($\sqrt2$) seems to have fractional form with a terminating decimal form, which (I think) is not true, then what is going wrong in the argument? Or else is it that the diagonal (in this case) has no starting and ending point?
Although I'm a little confused by your language, I would interpret this in one of two ways:
"Premises" 1 through 3 (they're actually axioms) redefine lines and points in $\Bbb R^2$ in a way other than the one we're typically familiar with. Under these rules, I would say that Premise 4 fails because the diagonal of a square is not a line, since it's end point violates Premise 1.
You're operating under a different metric. In other words, distances aren't measured in the the traditional way in your defined space. For instance, if you work in a 2-D metric space using the Taxi Cab metric
$d((x1 , y1), (x2 , y2)) = |x1 - x2| + |y1 - y2|$
then Premise 4 works out fine, although the length of the diagonal is actually 2 in this case, not an irrational value. Indeed, the length of the diagonal of a square under this metric (provided the square is parallel to the axes) is just the sum of the height and width of the square, so it's rational.
TL;DR
Either the length of the diagonal in this space is irrational, in which case the diagonal isn't a "line", or the metric is such that the length of the diagonal of a square with rational sides are never irrational.