my $compiled_regex = qr/$regex/;

my $sth = $dbh->prepare("insert into regexes name,value (?,?)");
$sth->execute("ready_to_eat", $compiled_regex);

# later

my $sth = $dbh->prepare("select value from regexes where name = ?");
my $rc = $sth->execute("ready_to_eat");
my @values = map { $_->value } $sth->fetchrow_array({}); # or something, look in the DBI docs

# convert back to compiled regex
for (@values) {
  $_ = qr/$_/;
};