This is O(n) rather than the O(n²) of your solution:

#! perl -slw
use strict;

my @strings = (
'string that I need to gather the common base from: number 1 and some 
+other junk in it',
'string that I need to gather the common base from: number 2 and some 
+other junk in it',
'string that I need to gather the common base number 4 and some other 
+junk in it',
'string that I need to gather the common base from: number 3 and some 
+other junk in it',
);

my( $mask ) = ( $strings[ 0 ] ^ $strings[ 1 ] ) =~ m[(^\0+)];
my $common = substr $strings[ 0 ], 0, length $mask;

for my $i ( 2 .. $#strings ) {
    if( substr( $strings[ $i ], 0, length $common ) ne $common ) {
        ( $mask ) = ( $strings[ 0 ] ^ $strings[ $i ] ) =~ m[(^\0+)];
        $common = substr $strings[ 0 ], 0, length $mask;
    }
}

print "'$common'";

__END__
C:\test>1147616
'string that I need to gather the common base '
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