Determining the correct depth of a well
The well needs to be far enough down to reach the underground water table. There may be certain requirements put forth by your local area (township, county, state), but other than that, there's not usually a "correct" depth. 

As Doresoom said, your best bet will be to ask the neighbors. They'll know whether there's enough water. Keep in mind, though, that with well water, there is always going to be a chance that you'll run out (if it doesn't typically, a particularly long drought might make you run out).

That said, it's not uncommon for people to build a second (or even third) well, if they have more than one deposit of water under their land. So I wouldn't discount the house just because of the well. Also, since the house is now within city limits, you also probably have the option to hook into the city's water supply (but this might not be the case for all of the houses you've looked at, so the other information may still be useful).

Also, there are a number of ways to help conserve water, or use different water sources. If you're the type that likes to water their lawn, or wash their own car, or keep a garden, you would probably benefit from installing rain barrels to catch the water run-off from the buildings on your property (if you use the water for anything that you will eat or drink, such as a food garden, make sure you have it tested regularly, and/or run it through your filter system). You just have to get creative. :)