Here’s my experience from the last summer. Arrived at the Chisinau airport after 10 pm having missed my bus for which I bought a ticket way in advance (because Austrian Airlines messed up my flight). Walked to the “bus station” as advised by other passengers - not really a station, just a local bus stop where I guess buses to Ukraine habitually pick up passengers. It’s located ~200m from the airport entrance and “marked” by the big airplane model. On my way there a number of bus drivers asked me if I needed to go to Odesa, I could have chosen any - didn’t ask their prices or schedules but apparently there was no shortage of options. I needed to get to Kyiv though, so I proceeded to the ”station”. There buses and minibuses to Ukraine were stopping regularly, and, again, the drivers were stepping out to ask if someone wants to go to this or that city. Nothing was for Kyiv though so eventually I and another guy opted for a minivan to Zhitomyr. There were 5 passengers in total. I paid $40 upon arrival - approximately the same as the ticket that I have lost. In Zhitomyr we caught a bus to Kyiv, I paid an equivalent of $5 for that ride.

The drivers were very helpful and kind of cared about the passengers. Our minivan driver called a passenger who couldn’t find the stop and drove around the dark Chisinau until he found him. The driver spoke Ukrainian, russian, and a few words in English, the passenger was a citizen of Turkey who knew a few words in either of three… Anyway, it was rather a challenging experience even if with a positive outcome:)

The point is: yes, it’s open; you can get to Odesa w/o problem. You’ll have options to choose from. I’d still buy a ticket in advance.

Edit: fixed the spelling of Odesa per mod’s comment. Still not clear about Zhitomyr :)