I am writing this to give you some idea on tenting around Odessa these days.
1. In Odessa itself.
I'd not recommend tenting in Odessa, as it is not throughoutly safe. But if you are brave - you might try it.

The red dot is monument to Shevchenko in Shevchenko park - our meeting place. Note the park that goes all along the seaside. Sometimes occational tents can be seen there, so you may try to place your own there too. Though, let me say it once again - I'd not recommend it.
2. Sukhoy Liman
In a village not far from Odessa on the shore of firth (swimming there is at your own risk, I'm not sure the water is very clean). I can't also tell anything about the behaviuor of locals - but many fisherment tents can be seen around, so I assume it's ok.

How to get there: you can merely take a bus that goes to Sukhoy Liman (Сухой Лиман) - depart from Privoz (Привоз) bus station which is right near railway station. Or take any bus that goes to Tavria V - Marshall trade center (Маршал) and then hitchhike.
3. Karolino-Bugaz - recommended option

While distant from the city of Odessa, it's a nice campling place on the sea shore. Nearby attractions include Belgorod-Dniestrovsky fortress and a shipwreck
How to get there:
Take a bus from the railway station that goes to Belgorod-Dniestrovsky (Белгород-Днестровский) or from Privoz bus station that goes to Zatoka (Затока). Get out at Karolino-Bugaz (ask driver or people) and go to the seaside. The road will take ~1 hour.
Hithhike via Ovidiopol road. To get to it's start, take any bus that goes to Ivanoskiy most (Ивановский мост) or Selekzionny institit (Селекционный институт).
Or take a local electric train that goes to Belgorod-Dniestrovsky (will be a good experience if you did not go by it before) from railway station. The road take ~1,5 hour, trains depart at 8.42, 11.39, 14.18
Bonus: a map of railway station and surroudings
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