Can we reconstruct/reset a broken Postgresql data directory without losing data? I'm trying to restore a database from a broken data directory that have lost many of its files accidentally.

I have tried unsuccessfully to install new instance of the same Postgresql version and then copy (with overwrite) the broken datadir to the new one.
here the remaining datadir that I'm working on :

   |-base
   |---1
   |---16384
   |---17950
   |---19621
   |---pgsql_tmp
   |-global
   |-pg_multixact
   |---members
   |---offsets
   |-pg_notify
   |-pg_stat_tmp
   |-pg_tblspc
   |-pg_xlog
   |---archive_status
It may be possible to rescue some of your data. It will not be straightforward.

Hopefully you took a full file-level backup of your postgres directories as soon as you noticed the problem. A recent filesystem backup will also he useful.

Then, subscribe to the postgresql-general mailing list and post a full description of your problem there. Bear in mind it is the weekend and you will need to be patient.