Restoring Postgres from only base backup taken with wal-e
I have a postgres base backup taken via wal-e.


So I would suggest that your question would be best addressed to the people who created wal-e.  I've never used it myself, but ... 


  Is it possible to start the DB and bring it to consistent state with only the base backup?


Short answer: No.  

The base backup takes a "snapshot" of the database. 
That snapshot, taken from a running database, it unlikely to be consistent in itself. 

Anything and everything that happens in the database after that snapshot is taken is written to the WAL files so, in order to do any meaningful database recovery, you need those as well.