It is odd that you highlight the Bedford Level Experiment, rather than other methods that have been used for thousands of years. The new experiment has the advantage that it can be performed by a single person in a single afternoon. It has the disadvantage that it shows that the Earth is flat.
Eratosthenes measured the north-south curvature of the Earth by making observations separated by hundreds of miles. It could be applied east-west with good clocks, or, as you suggest, with the simultaneity of telephones. Since I'd have to travel hundreds of miles anyway to reach the straight canal in Bedford, it has little advantage over Eratosthenes's method. I suppose you could make a similar observation by climbing a mast on a ship the right distance from shore, but the ocean waves add noise not present on the canal. It does have the advantage of requiring less geometry. Since the Bedford experiment used 1/100 the distance, it required 100x the accuracy of angular measurement. This is easy to overlook, since the measurement is not phrased that way, but I think this is why it encounters new sources of error.
Older experiments are generally easier. While everything is easier to measure today, the main advance is in measuring time.
Bedford Level Experiment [...] has the disadvantage that it shows that the Earth is flat.
I love this. As it happens, I live quite near Bedford and am terribly tempted to actually try it one day. (Edit Looking closer, turns out the Bedford Level is in Norfolk, not Bedfordshire, so a little less nearby than I thought.)
There are loads of fun ways of verifying that the Earth isn't flat. Some of these were easily available to the ancients - e.g. the shape of the shadow of the Earth on the Moon during a lunar eclipse (it's always a curve). Others are easier ...
Suppose you distrusted everything you had ever read about science. How much of modern scientific knowledge could you verify for yourself, using only your own senses and the sort of equipment you could easily obtain? How about if you accept third-party evidence when many thousands of people can easily check the facts?
My purpose with the question isn't to cast radical doubt on science; rather, it's an entertaining game of trying to understand how we know what we know. Thinking through these sorts of questions also helped me notice interesting things in the history of science that I hadn't previously focused on. It might also be of interest from a science education perspective.
Some things are much easier to check than they used to be. As late as the 19th century, there were people who were publicly skeptical about the curvature of the earth. Skeptics and scientists did careful measurements (notably the Bedford Level Experiment) to observe the earth's curvature. Today, you can verify it by phoning a friend a few time zones away and noticing that the sun reaches the zenith at steadily later times as you move west. This only makes sense if the earth is curved.
Some things are still hard to check. I don't know an easy way to show that the Earth orbits the Sun. The direct way to show it would be to measure stellar parallax. But even the closest stars have a parallax of less than an arcsecond. My understanding is that very few amateurs are able to take measurements with that level of precision.
Some things are surprisingly easy. There are lots of easily accessible demonstrations of quantum phenomena. For example, a ten dollar spectroscope will show you that an incandescent light bulb has a continuous spectrum, and that LEDs and fluorescent bulbs don't. Bright-line spectra are very much a quantum mechanical phenomenon -- it's a sign that the atoms in the light source have fixed energy transition levels. Spectroscopy was one of the key early lines of evidence for quantum mechanics, and it blows my mind that it's something you can just see whenever you want, with a negligible equipment cost.
Pretty much all of modern chemistry and solid state physics rests on a quantum foundation, and you can test a great deal of chemistry pretty easily. If you are in doubt that water is a bonded compound of two gasses, you can do the electrolysis very easily yourself. You can observe the periodicity of chemical elements yourself if you buy alkali metals (don't try this one at home!). If you are willing to accept slightly indirect evidence, the entire semiconductor industry is about precisely controlling the conductivity of impure silicon, and this would make no sense if quantum mechanics weren't a reliable guide to electron energy levels in the solid state.
I don't feel quite as qualified to play this game for biology. I imagine that antibiotic resistance is a well-enough documented case of evolution through natural selection to serve at least as a proof of concept. DNA sequence comparisons across species are emphatic evidence of taxonomic trees, if you trust the scientists not to be part of a vast conspiracy.
It feels almost impossible that it's easier to see quantum mechanical effects than it is to see that the earth orbits the sun, but it does seem that way.
Some questions: