How can I find whether there is oxygen present inside a well of diameter 1 m at 50 m's deep?
The ideal solution would be a single gas personal detector, such as this one (it's only the first example I found, I don't know this brand, but there are many equivalent apparatuses from other vendors).

While digital personal detectors are expensive ($200+ range), there are simpler systems which also provide some basic safety (better than nothing). The Davy lamp is perhaps the best known example of these, and it was a huge breakthrough in mine safety when it was popularized after 1815.


(Source: http://lindal-in-furness.co.uk/MinersLamps/MinersLamp.jpg)