Suppose H is actually the true height of the building

Your calculated value could be 5% high or 5% low.

If h is 5% higher than the true height (H) of the building then

h = H + 0.05H

h = 1.05H

h/1.05 = H = 104.6657

or

If h is 5% lower than the true height of the building then

h = H - 0.05H

h = 0.95H

h/0.95 = H = 115.6832

The angle of elevation must be between:

Angle1: arctan (115.6832/40)

&

Angle2: arctan (104.6657/40)

The question is "How accurately must the angle be measured..."?

Determine half the difference be Angle1 & Angle2.

That is how accurately it must be made.

Most likely you are required to get the angle to the nearest tenth of a degree.

or

it could be plus or minus minutes of arc.

.