Excite me a wave which is exactly square.
A square wave basically plots between a low point and a high point, only - it is a non-sineoid waveform, according to wikipedia. So... binary? Plot your ones, and your zeroes, in a scatter-plot. Then, connect the dots - you know it has no in-betweens, so straight up-line form zero to one and straight down-line from one to zero, all on a forward line for time. If your computer is being all picky about lag-time and the time spent on rising and falling, tap it firmly and plot the silly thing by hand. A perfect square wave has an equal number of highs and lows (50% duty cycle) so make sure your binary message is long (or short) enough, or specially formulated enough, to get a perfect square wave. Or else "test plot" just alternating ones and zeroes.
Draw me an exact circle.
Take an eyedropper and a container of pure water. Carefully hold the eyedropper straight, and let a single drop of the water fall. While it is falling, its circumference at the equator should be an exact circle, since water's cohesion should pull it together in a sphere. Granted, it's hard to measure - but the circle has been created, and drawn with a description rather than a line. For an alternate method, build a space shuttle (or talk to those who have one) and repeat the experiment in space, where the shape of the water should form and exact sphere.
Show me a container that is exactly full.
Take a solution of distilled water and common dish soap, in a 6-to-1 ratio (optional: add glycerin or corn syrup). Mix well. Take a straw, a length of pipe, or a bubble wand, dip the end in your solution so that a thin film covers it, and gently blow through the hole to blow bubbles. They are a container, check. They are full (of air) also check. No more air can be placed in them, and none taken out of them, or they will pop.
Take a sponge, and set it in a bowlful of water. Let rest until fully re-hydrated (some squeezing and expanding may help). The sponge is now exactly full of water. The fact it can be taken out of the water, and still be waterlogged, makes it a container. If you take it out of water, without squeezing, it may still be full of water (perhaps exactly so for a little bit) - but it's harder to judge the "exactness" of the fill or exactly when it is done dripping.
Bind me a book which is exactly endless. (Reader will always read left to right, top to bottom, one page after another)
Start with a standard hole-punch, and the correct number of binder rings (most hole-punchers make three holes, some do four). Binder rings are effectively smooth (though ones with as little difference in the hinge and clasp are preferred), so any pages, when turned, will quickly circle around to land at the back of the sheaf of papers. To make "truly" endless, add increasing numbers of pages until they have comfortably filled the ring, and the book can't "close" to a single sheaf but must remain standing, with the pages equally splayed out around the circumference - it can still be read, since even when full at the ring site, the edges are quite loose and can be turned. To make it a book, fill the pages with something, perhaps art and complex images, so one can continue revisiting the pages indefinitely, rather than a simple story which might not make sense out of order or may not be worth rereading.
Make me an apparatus which rotates exactly once per year.
Making a clock-like mechanism shouldn't be too hard. Make sure it's sturdy enough to last at least a year (under any conditions), and also - instead of rotating a hand, have the motor rotate a disk set atop another disk, just for aesthetics. Have on top of the second disk, another rotating mechanism that can be set at an angle. Take it and run away - north, or south, depending on your location and travel habits, but you need to get over one of the poles - and I mean right over the turning point, not the magnetic pole or what have you. Set your apparatus down right over the pole (consider setting up a shelter over it, you built it study but why take chances), and set the mechanism to turn once per day - clockwise, exactly one rotation per 23 hours, 56 minutes, and four seconds. The earth rotates counterclockwise at the same rate, bringing the rotating disk of the apparatus to a halt regarding the earth's daily rotation. The second mechanism should be set to one rotation per year, and the mechanism tilted by 23.4 degrees - this should counterbalance the earth's tilt. In exactly one year, though, it has rotated exactly once around the sun - the rotation will occur at exactly the one year mark because one rotation about the sun is the actual definition of a year, it is 365:5:48:45 on average. Larger rotations, like the sun's or galaxy's rotating, are so immeasurably large, and the portion our apparatus got through in a year so small, that they don't count - that is, exactly one sun-rotation, and some extra wobbly non-rotation distance moved.
addition: Because the axial tilt is a genuine pain, take our clock-mechanism, head over to the pole, and build a tiny railroad in a complete circle around the pole. It needs to be exactly 1,618.60075 miles long, which means 257.6085 miles from the pole in every direction (you may want the south pole for the extra room). Set your device on a tiny train on the tiny railroad, set it going counterclockwise with a speed of 3.25 inches per second - which will really cancel the axial tilt, and prevent a small second rotation from being traced by the device spinning around the tilt-axis.
go back to the folk you got your space ship from back in drawing an exact circle, and borrow a space ship for a brief jaunt over one of the poles (and their computers) and plant the device...in space, in a reverse-geosynchronous orbit, directly over the axis perpendicular to the earth's rotation (that means, 23.4 degrees from the pole) or else in a sun-synchronous orbit over the equator. Use the two rotating mechanisms to cancel out any other rotations you would like to eliminate (the first will cancel the tilt but the daily rotation will need countering, the second will cancel the daily rotation but the tilt will need countering, and one to spare if the satellite it's attached to might rotate and that needs countering, whatever).
Find me two objects of exactly the same color.
Take two samples of a pure substance - composed of a single element or reliably pure compound. It is important that the substance be pure, 100%, but not really what the substance is - even distilled water will do as long as it is pure. Since we want two objects, you could even purify the samples from different batches - and for potential bonus points, have them identically shaped or held. Take the samples into a lead-lined darkroom, set them on a table, and shut off the lights. Your objects are now exactly the same color. Since they are a pure substance, no chemical test can distinguish between the wavelengths of light they will reflect, and in such perfect darkness, your eyes (or anyone else's) can't be fooled by apparent differences brought on by exact shape, any containers, or specific placement in regards to the light source (angles or shadows).