We just played with Elastic File System mounted into a Pod via `nfs` and it worked like a charm, with the additional "oh, wow" of being able to attach the same EFS to several Pods at the same time. I was also thrilled that they mounted with the root uid intact so there wasn't any kind of dumb permission juggling.
I did say "played with" because we haven't beat on it to know if one could run Jira, Gitlab, Prometheus, that kind of workload. I wouldn't at this point try Postgres but maybe it'd work.
I wonder how suitable EFS is for Postgres. It's supposedly low-latency, high-throughput and supports Unix locking semantics and so on. On the other hand, it's NFS (one of the worst protocols out there), and there have been reports of less than impressive latencies. EFS is also a lot more expensive than EBS.
There are still some bugs with this, particularly on AWS. Getting better with every release, though.