I have the following output from SAS for a data set I entered, using a linear model.
From what I have learned, I made the following two conclusion:
The residuals do not lie in a band around zero, so the error terms do not have constant variance, and the curvature also suggests a lack of fit of the linear model.
A lack of fit test (done through SAS) gives me a low $p$ value, which indicates that SLR is not a good fit. However, a Brown-Forsythe test yields me a high $p$ value, suggesting that constancy of variance is a reasonable assumption.
But I do not understand how this picture depicts constant variance.
Any help much appreciated.

First of all, I would look into the two drastic outliers (the point in the top right corner and the one around length $l = 130$). I would strongly consider removing them if you think that you have a proper reason to do so. Investigate why they are like that, and be sure to document this.
Second, back to your question, I would look at a predicted length vs. studentized residual plot. See if all of the residual points (minus the two I specified before) are scattered both above and below the axis.
Removing those points would likely improve the fit of your model.