All are correct, technically speaking, but #2 doesn't make sense. That sentence is implying that many Russian families share one dacha! I don't think that's what you mean.
The others are all fine.
1. A lot of Russian families have a dacha.
2. A lot of Russian families have their dacha.
3. A lot of Russian families have dachas.
4. A lot of Russian families have their dachas.
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Which ones are grammatical? Do they have the same meaning?
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