Earlier this year I had a chance to take a photo of the "blood moon" - the eclipse of September 27, 2015. I used an "ordinary" camera: a Panasonic FZ200 with 25x optical zoom, without a tripod but with image stabilization.
When I looked at the pictures on a big screen, I noticed a few little dots in the background which I initially assumed to be noise (or defective pixels). But then I pulled up the "Sky Guide" app for that time and location, and discovered that I actually had accidentally found some faint stars that were visible because the moon was so dim.
So I decided to see what would happen to the apparent location of the moon relative to the background stars if I programmed a different location on earth. To make it a little bit realistic, I chose a point that was about 200 miles (300 km) away - New York City. And then I drew a few lines...:
On the left you see the image from the Sky Guide app (I had to rotate it slightly to align with my picture) and on the right, the photograph I took. The intersections of the two sets of lines through the most visible stars falls just inside the disk of the moon. I circled the stars in my photo because they are quite faint.
Next, I decided to compare the image I obtained for a location close to where I took the photograph, "Saratoga Springs, NY" and a location about 300 km away, "New York City, NY" at the exact same time. I got the following:
The dimensions given are in inches on the screen: the diameter of the moon is 1.67", and the two green lines are 1.10" and 1.46" respectively. Since these lines intersect the edge of the lunar disk at a fairly acute angle, we can estimate their distance quite accurately from these measurements. Unfortunately, for the other direction the lines are quite close to the center of the disk, so it is less easy to measure their distance accurately. However, we can use the line of intersection of the vertical diameter and the diagonal to get "somewhat close".
When I put these two sets of lines on the same image, you get a measure of the displacement of the moon relative to its surroundings (background):
This displacement is approximately 0.25", or 15% of the diameter of the lunar disk. Since the moon subtends an angle of approximately 0.5° relative to Earth, this means that the angular displacement of the two points relative to the moon is about 15% of that, or 1.3 mrad.
We can now estimate the distance between these two points - it should be the distance from the surface of the Earth to the moon, multiplied by that angle. If we say the moon is about 400,000 km away, then we get a distance of 500 km. That's about 60% bigger than the actual distance of 300 km; this shows that it's hard to get an accurate measurement using only the tools you specified.
Note also that the moon and stars move across the sky pretty rapidly: to cover 360 degrees in 24 hours or 15 degrees per hour, it moves about one lunar diameter every four minutes.
This means that you need to know the time of the photograph quite accurately if you want to estimate the position of the camera - the error I had above (200 km) corresponds to about 40 seconds (google "the problem of longitude" - this has bothered seafaring nations until the advent of GPS). I had the benefit of using software that could observe the moon "at the same time" at two different locations - usually life isn't that accommodating.
So - assuming that you have an accurate clock, and take the photo of the moon at the time of a total eclipse so you can observe near-by stars, a hand held camera can get your position to within 100 miles or so. Note that I am assuming you can't do a good measurement of the angle of the moon relative to the horizon (the usual trick with a sextant... it uses mirrors to project the image of the moon onto the horizon for an accurate measurement. If you try to do that with a camera, the angle will usually be too large to be measured accurately, unless the moon is really close to the horizon).