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I would like to style a layer where the fill of the symbols should be graduated e.g. based on an attribute showing the "proportion of the female population" and the size of the symbols based on another attribute e.g. "total population size".

In -> Layer Properties -> Symbology -> Graduated I can choose as a "method" either "size" or "color" (QGIS 3.2.2). However, I would need both.

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You can always define an attribute based size in the Symbol Selector dialog (click Symbol | Change...), using the Data defined override menu; check the image below:

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The Edit... option opens the Expression String Builder; select your desired field in the Fields and Values tab in the middle panel and hit OK. Alternatively, you can add math and/or functions to refine the size.

This is independent of any settings concerning the Graduate styling.

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    Just to expand on that, it's also useful to click the "Assistant" option in that menu, which will provide an interactive way of setting the property variation – ndawson Sep 4 '18 at 1:39

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