Can factors be significant if there is a lack of fit and weak R²?

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Your assessment is essentially correct. Your one factor that was identified as significant probably is significant. The model, however, is not a good model. As you speculated, there are a number of reasons for this such as missing factors, interactions, quadratic terms, or a covariate. Also look at your data to see whether you have a few data points with large residuals. Investigate and validate these as correct. Adding the correct missing factors should improve your R². If you address the lack of fit and find that you still have a poor R², you then need to investigate the sources of your experimental error starting with the measurement system.

 

What Miner said. Plus remember that the R squared value is essentially a measure of the amount variation due to the factor(s) being evaluated. In other words a low R square means that most of the variation is driven by one or more factors that were present in the study but not tested for. Also a very important thing that is too often overlooked is that statististical significance (no matter how low the p value is) does not mean practical importance. In other words the p value does not tel you if the identified factor is the dominant or controlling factor. To have a useful model you must have identified the controlling and occasionally strongly contributing factors.

 

Some experiments can best be described per the following quote by Andrew Gelman:

"My best analogy is that they are trying to use a bathroom scale to weigh a feather—and the feather is resting loosely in the pouch of a kangaroo that is vigorously jumping up and down."