Relationship variance, accuracy, and precision? I have built a measuring machine. It takes a photo of something and tells you how tall that thing is.
I took three photos of the same object. On those three occasions, my machine said the object measured the object to be (in meters):
0.359595
0.335077
0.318732
I have computed the variance to be 0.000423013. The mean is .3378.
If I do not know the "true" size of the object I am measuring and assuming no other information is available, what bounds can I place on the accuracy and precision of my measuring machine?
 A: Without knowing the true value, I don't see how you can speak to accuracy for a given height estimate.
The variance component could be defined as the imprecision of the estimate (whether or not that estimate is correct).
That said, you surely have trained your algorithm using some known values or truth, such that you could at least provide the accuracy and precision for the training data.  This will obviously differ, but at least there is some information about how well it performs.  If you used cross-valiation and test samples, all the better in providing how well the algorithm performed overall (e.g. precision, recall, accuracy, efficiency etc.).
A: I will try to complete the good answer from Minnow. Let's start from scratch.
Your problem is simply a regression problem i.e. you found a function $f()$ such that it gives you an output $y_i$ (height) for an given input $X_i$ (image). 
The first step is to train or build your model. You are using some model (based on physics) or some data $\{X_i,y_i\}$ for i in $[1..N]$ that you know to be correct. This step in your case is probably very simple, maybe using trigonometry on your image. However this is happening, you do know some correct values! At the end of this step you now know the relationship between $X_i$ and $y_i$ i.e. you know $f()$ such that $y_i = f(X_i)$.
Here you can feed your function whatever image you want and it will spit out a prediction but you will not be able to describe how precise or accurate is that prediction. You don't have any knowledge that your prediction is better than random. You can guess (and I think this one of the thing you are doing) that this is better because you have train/build your model for that purpose but you can not confirm it or be sure of it.
Precision and accuracy are about comparing model's predictions with values that you know to be true. Computing them is exactly the step that allows you to gain information about the fact that your model's prediction will be "more correct" than other predictions (in your case 0.33 is more likely to be true than 0.03). And then you will be able to extrapolate to new measurements.
Thus, the second step : measure how good is the function that you build previously. This require the knowledge of the true value, maybe you can compute that value using simple physics or trigonometry again but you need to know the correct answer. So you take some image $X_i$ and the correct height that you have measured $y_i$ then you use your model to get an estimate $\hat{y_i}$. You do that for a certain number $N$ of image and then you can compute the accuracy and precision (I recommend that you read this for the relationship of variance, accuracy and precision https://en.wikipedia.org/wiki/Accuracy_and_precision). You can compute accuracy using, for instance MAE and your 3 measurements (more than 3 would be better):
$$
acc = \frac{\sum{abs(\hat{y_i}-y_i)}}{N}
$$
This will be >0, then your model is not perfect ! And you can say, that on average your prediction is off by a certain amount equals to $acc$. 
I hope that all this make sense.
Let's go back to your example, you have one new measurements (or 3). If the accuracy is 0.003 then indeed you can say that this is very likely that the true value is somewhere in 0.33-0.003 and 0.33+0.003 thus more likely to be 0.33 rather than 0.03. However if your accuracy is 0.66 then the true value is very likely to be between 0 (-0.33 doesn't make for height) and 0.99. In that case you don't have a good knowledge of what is the true height.
A: In metrology "accuracy" is a qualitative term, i.e., it is not associated with a number. Thus, we can speak of one measurement technique being more accurate than another, but we don't say one method is 2 times more accurate than another. To do so would require knowledge of the true value, and if you knew the true value then we would not be talking. I think what you want to know is the combined uncertainty in a measurement made with your machine. Chances are pretty good you are going to have to calibrate your machine with standard lengths that have known uncertainty and propagate the uncertainties assuming one or more different probability distributions. In other words, you are not going to be able to do what you want without additional information/assumptions. I suggest that you look at the Guide to Uncertainty in Measurement to get more details about this process, which can be very non-trivial.
