Why doesn't acetone have high basicity?
The C−O bond is polar, O becomes partially negatively charged and it should attract protons, becoming a strong base.
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I don't know the reason, but I can describe the trend (source: http://cactus.dixie.edu/smblack/chem2310/summary_pages/pka_chart.pdf):
Protonated ketones have a pKa of about -7. Protonated ethers have a pKa of about -4 and protonated primary alcohols of about -2. Comparison to hydronium in water is difficult, but in DMSO, hydronium is more acidic by about 2 pKa units (source: comment by DavePhD below https://chemistry.stackexchange.com/a/27278/72973).
So it seems water is a little bit basic. If you replace one of the bonds to hydrogen with a bond to carbon, it becomes less basic (alcohol). If you replace the other bond to hydrogen with a bond to carbon as well, it becomes even less basic (ether and ketone).
A hand-waiving explanation would be to say the electronegativity difference between hydrogen and oxygen is higher than that between carbon and oxygen. I am not sure what to say about the difference between ethers and ketones. I think of ketones as the better hydrogen bond acceptors, and as more soluble in water (all other things being equal), so I don't have a good rationale why it would be easier for an ether to accept a proton.