No, there is absolutely no need for a fire engine, and it's perfectly OK to land at an unlicensed strip on a training flight with the owners permission and if the instructor is satisfied that the risk is acceptable.
I don't know the specific rules, but I wouldn't have thought this was a good idea (or permitted) for a solo student though
For a non-training flight, you can do whatever you want. Land in a farmer's field if you wish!