Apache 2.4.7 403 Forbidden Error

Here is how I solved this problem for anyone who might have a similar one:

It seemed the problem came from where I installed the overpass installation ($EXEC_DIR).

So I had to change the install directories to:

$EXEC_DIR       /var/www/osm/
$DB_DIR         /var/www/osm/db/
$PLANET_FILE    /var/www/osm/planet.osm.bz2
$REPLICATE_DIR  /var/www/osm/rep/

Giving the resulting default.conf:

<VirtualHost *:80>
    ServerAdmin webmaster@localhost
    ExtFilterDefine gzip mode=output cmd=/bin/gzip
    DocumentRoot /root/osm-3s_v0.7.4/html

    # This directive indicates that whenever someone types                         http://www.example.com/api/ 
    # Apache2 should refer to what is in the local directory         [YOUR_EXEC_DIR]/cgi-bin/
    ScriptAlias /api/ /var/www/osm/cgi-bin/


    # This specifies some directives specific to the directory: [YOUR_EXEC_DIR]/cgi-bin/
    <Directory "/var/www/osm/cgi-bin/">
            AllowOverride None
            Options +ExecCGI -MultiViews +SymLinksIfOwnerMatch
            Order allow,deny
            Allow from all
            #SetOutputFilter gzip
            #Header set Content-Encoding gzip
    </Directory>

    ErrorLog /var/log/apache2/error.log

    # Possible values include: debug, info, notice, warn, error, crit, alert, emerg
    LogLevel warn

    CustomLog /var/log/apache2/access.log combined

</VirtualHost>

I tried and tried but couldn't get it working with the old dir. Good luck!