What happens if you interrupt a POSTGRESQL database that’s receiving data?

I have a Docker application and as a quick and dirty backup strategy, I’m thinking to tar the whole project incrementally and send it off remotely.

I’d rather keep the services running while doing the tar operation.
The question arises, what happens if a client is writing data in the moment of tarring?

Will I have an incomplete or corrupt data entry?
If so, how much of a problem is that if you don’t care about that data being lost?

The other strategy is to stop the docker container and then do the tar.
In that scenario, I still wonder what happens if somebody is writing data at the moment of shutdown.

It seems like both scenarios might suffer the same problem and if that is so, I might just as well leave the service running during the backup.