Okay, so I may be wrong in my observations, and as there is no way for me to get Link to stand directly against the wall, I will put a range for the height instead of an exact answer.
As you can see here, Link is standing next to the meter right after he stretched and is almost standing straight up. It's a bit slanted, so Link's head is probably just under the mark, rather than right on it. His distance from the meter also means he is still a bit too tall, however, the meter is also a bit incorrect with its bottom mark being slightly above the ground, putting Link a bit higher on the scale. Link's shoulder width is also wider than his hips, so his distance may not actually be as great as it seems. His head is slightly slanted, his neck is forward, and his knees are slightly bent, so more height can be added in, but not a lot. Taking all of this into account, he's probably at 1.69 m (5'6.48") or 1.7 m (~5'6.9") in this picture.
I actually took a lot more pictures, but I can't upload all of them, since they ended up being too large. In one of them, I got link to stand beside the meter, and he stood at just over the 1.6 m mark; it looked to be at 1.61 m (5'3.36") or 1.62 m (5'3.7"). In fact, all of the times I took picture of him standing beside the meter (without him stretching), he stood at around 1.6 m (5'3").
If you are wondering why these are in meters, the scientist in the lab tells you how many meters you go down whenever you dive into the water. This is because he was watching you, according to his dialogue. With the ruler being in the water and the dialogue only giving the dive lengths in meters, we are led to assume the lines are meant to be in meters.
Personally, from my own observations, I'd say Link is somewhere between 5'3" and 5'6.9". I'd like to believe he's 5'4.5" or 5'5", if not 5'6", but I can't be sure. If I could get clarification from a developer of the game, I'd be more confident in my words, but I can't actually do that. Who knows? I could have measured wrong. Maybe Link is actually under 1.6 m. In any case, if you want better information, you could try playing the game and looking at it yourself. It is really easy to do this if you wear the Zora's Tunic since you'd be able to stay underwater.