I have seen several videos and tutorials regarding height gauges but I haven't come across one that show the measurements in 0.1mm or better.
Can this be made with arduino?
Accuracy: 0.1mm or better
Range: 0 - 150mm
Why? You can buy a digital height-gauge scale with a data output.
An Arduino could read just about any suitable sensor.
Noncontact distance sensors with accuracy that high tend to be quite expensive. However, cheap digital calipers are popular. What did you have in mind?
If this is for measuring the height of people, then you are asking for an accuracy less than the thickness of a hair.
You will need to amplify that 0.1 so you can measure the even smaller gradient between 0.0 and 0.2.
Yes!
The real answer depends on the sensor you use. The sensor will determine the measurement range and accuracy, not the Arduino.
Please post a link to the sensor you are considering and explain what you want to measure. Without that information and the other project details, any answer would only be a guess and would probably be wrong.
The real question you need to ask is: can I make a height gauge with 0.1mm accuracy or better. The Arduino or any attached computer/display has absolutely no bearing on accuracy. It is merely the sensor and display device. Accuracy must be designed/built into your mechanism.
That will be very short people, OP specified 0..150 mm 
I have used Mitutoyo callipers at work. These have the range and accuracy you need but (a) they are expensive and (b) they have a proprietary data output that needs various special cables and transmitters which may take a bit of work to interface to an Arduino.
There are probably cheap clones on AliExpress.
I know there are cheap ones too, but I wanted to do something using arduino so I can continue to learn. I do not really need any data output other than a digital readout.
I guess the better way is to just get that digital height gauge (intended for a thicknesser machine). So much upside like being cheap, less work and accurate to 0.01mm.
Thanks for all the replies...
I think, the best answer to your question wrote @gilshultz.
Anyway, I am curious about this need for better accuracy than 0.1mm. As I know from your other topic, it is for woodworking machine.
I have some experience in this field. I'm making furniture as hobbyist. In my opinion, accuracy 0.01mm is nonsense. Most time, you do not have the planner blades sharp enough for this, plus you need keep in mind the vibrations of the material and the machine.
Even with the sharpest blade and the machine with high stiffness there will be a little fuzz and dents on the surface. 0.1mm accuracy seems enough.
I would be happy with finished timber dimensions within a millimetre of the intended size, say for shelf type projects. Small scale stuff like picture frames and possibly high end/exhibition furniture projects would have to be better than that.
A lot of timber (natural) is going to move depending on temperature and moisture, maybe not so much with manufactured materials like MDF. The UK heatwave is playing havoc with external timber. Nails are extracting themselves with the contraction and expansion.
I use digital vernier callipers a lot, mainly the 150-mm type. Occasionally, traditional verniers and dial gauges. Often thought about hacking one of the digital types out of curiosity.
A dial gauge is probably more hackable with some sort of encoder on one of the dials.
while most of my work doesn't need this kind of accuracy... the machine is a pantorouter and I need it to place the mortise and tenon joint on the center of several pieces. Not near center, not close but centered as the matching piece would or could misalign. Quality of work is often a matter of preference, some would think that a slight misalignment is fine since there is wood movement. But having very demanding clients in the audiophile industry, I chose to produce works that receive zero bad remarks.
You might be interested in a video made by someone who makes very expensive turntables.
By expensive, I mean £30k, bespoke.
Off the top of my head, it might be the Yorkshire Woodworker or similar.
Oh, I would love watching them. Post if allowed. Although I am not making turntable plinths, I still enjoy fine woodwork. And in the audio industry, £30k is somewhat normal... lol
Search for The New Yorkshire Workshop.
The one I watched was for a £13k turntable, but it was birch ply, laminated.
You'll probably like the final finishing, piano gloss.
There's something vey satisfying about listening to music with top quality equipment.
The guy who does the work, just gets on with it, little or no dialogue.
I've a lot of admiration for people at the top of their trade.
I have actually come across that video a long time ago... what I admire is his veneering work.
Sure.
5 microns!
Thanks Jim... plenty other choices that are cheaper and complete already that is perfect for my need. No electronics and coding needed.
Planer Digital Readout
The title of your topic said "with Arduino", so I was looking for solutions you could use with an Arduino.
You should marke the topic as solved, so that others don't waste their time looking for an Arduino solution.
That is why dial indicators are used! I built a height gauge using one to measure the height of ICs on circuit boards by a pick and place machine. Bought a used digital dial indicator, but never added the external display. The tech running the machine decided it wasn't that important. It has a heavy brass base. Janitor's granddaughters liked to play in the plant while Grandma was cleaning. They dropped it and broke it.