A compound need not "contain ions" to be polar. Any compound with molecules with centers of positive and negative charge that don't coincide is polar. If the molecule is not symmetrical, and contains atoms with a wide range of electronegativities, it's likely to be polar.
Table sugar (sucrose) is a polar nonelectrolyte. Sucrose is quite soluble because its molecules bristle with water-accessible OH groups, which can form strong hydrogen bonds with water. So sugar is not an exception to the "like dissolves like" rule of thumb.
Nonelectrolytes can be either polar or nonpolar. Nonpolar nonelectrolytes can be sparingly soluble in water, especially if they contain small electronegative atoms that water can hydrogen-bond to. Examples are O2 and CO2. If these nonpolar substances didn't dissolve in water, life as we know it would not be possible!
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