Can machine learning algorithms be trained to discard nonsense?
The problem you're describing can perhaps be treated as "anomaly detection". Google the term in quotes, you'll get a ton of references. So the idea would be to first train an anomaly detector to separate the "apples, oranges, and other fruits" from non-fruits (or "nonsense" in OP terminology). Then train a classifier on the fruits.

Update. In light of additional answers -- and too long for a comment.
The problem with declaring a "non-fruit" category highlights the distinction between classification and anomaly detection. In classification, we assume 2 or more coherent classes. Apples, oranges, bananas -- these all have well-defined common characteristics. On the other hand, "non-fruit" is not a coherent category. It can only be meaningfully defined as something like "not fulfilling enough of the fruit characteristics". Anomalous objects are not a coherent class; they are by definition anything that is not sufficiently close to a "normal" object. We do not expect examples of anomalous objects to be informative for training. After all, you can just generate random pixel-images and get tons of anomalous examples for free!

This is why anomaly detection requires techniques distinct from classification.