How can I parse through this?

There are many ways you could parse your file. Here is one way:

#!/usr/bin/perl
use strict;
use warnings;

my $section = "unknown";

my %volumes;    # volumes for each host, keyed by host name
my %capacities;  # capacity of each volume, keyed by volume name

while(my $line = <DATA>) {
    chomp($line);
    if ( $line =~ m/^host_output$/) {
        $section = "host_output";
        my $discard = <DATA>; # discard the header line
    } elsif ( $line =~ m/^volume_output$/) {
        $section = "volume_output";
        my $discard = <DATA>; # discard the header line
    } elsif ( $line =~ m/^$/) {
        $section = "unknown";
    } elsif ($section eq "host_output") {
        my ($volume, $host) = split(/\s+/,$line);
        push(@{$volumes{$host}}, $volume);
    } elsif ( $section eq "volume_output" ) {
        my ($volume, $capacity) = split(/\s+/,$line);
        $capacities{$volume} = $capacity;
    }
}

foreach my $host (sort keys %volumes) {
    foreach my $volume (@{$volumes{$host}}) {
        print "$host: $volume: $capacities{$volume}\n";
    }
}

__DATA__
host_output
volume_name   host_name
vol1          host1
vol2          host1
vol3          host1
vol4          host2
vol5          host2
vol2          host2

volume_output
volume_name size
vol1        10g
vol2        20g
vol3        30g
vol4        30g
vol5        20g
[download]

Which produces:

host1: vol1: 10g
host1: vol2: 20g
host1: vol3: 30g
host2: vol4: 30g
host2: vol5: 20g
host2: vol2: 20g
[download]

This example makes several assumptions about your data, which you should consider carefully.

Update: removed extraneous "next;"