Solved… in part.
Apparently, Installing the latest versions of OS X (e.g. Yosemite or El Capitan) removes some directories in /usr/local/var/postgres.
To fix this you simply recreate the missing directories:
mkdir -p /usr/local/var/postgres/pg_commit_ts
mkdir -p /usr/local/var/postgres/pg_dynshmem
mkdir -p /usr/local/var/postgres/pg_logical/mappings
mkdir -p /usr/local/var/postgres/pg_logical/snapshots
mkdir -p /usr/local/var/postgres/pg_replslot
mkdir -p /usr/local/var/postgres/pg_serial
mkdir -p /usr/local/var/postgres/pg_snapshots
mkdir -p /usr/local/var/postgres/pg_stat
mkdir -p /usr/local/var/postgres/pg_stat_tmp
mkdir -p /usr/local/var/postgres/pg_tblspc
mkdir -p /usr/local/var/postgres/pg_twophase
Or, more concisely (thanks to Nate):
mkdir -p /usr/local/var/postgres/{{pg_commit_ts,pg_dynshmem,pg_replslot,pg_serial,pg_snapshots,pg_stat,pg_stat_tmp,pg_tblspc,pg_twophase},pg_logical/{mappings,snapshots}}
Rerunning pg_ctl start -D /usr/local/var/postgres now starts the server normally and, at least for me, without any data loss.
UPDATE
On my system, some of those directories are empty even when Postgres is running. Maybe, as part of some “cleaning” operation, Yosemite removes any empty directories? In any case, I went ahead and created a ‘.keep’ file in each directory to prevent future deletion.
touch /usr/local/var/postgres/{{pg_commit_ts,pg_dynshmem,pg_replslot,pg_serial,pg_snapshots,pg_stat,pg_stat_tmp,pg_tblspc,pg_twophase},pg_logical/{mappings,snapshots}}/.keep
Note: Creating the .keep file in those directories will create some noise in your logfile, but doesn’t appear to negatively affect anything else.