$tables{$day}->{$state} = \[$deathsToday, $m, $double, $intercept, $corr];

This is setting that hash value to a reference to a reference to an anonymous array, not just a typical reference to an anonymous array ("arrayref"). Is that what you want? If not, you can drop the \ and access the first element via $tables{$day}{$state}[0] and the whole array via @{$tables{$day}{$state}}, or, on Perl 5.24 or higher*, $tables{$day}{$state}->@*. Note the -> operator can be dropped between {}'s and []'s.

Otherwise, if you want to stick with your current data structure, to access the first element you'll have to say ${$tables{$day}{$state}}->[0], or, on Perl 5.24 or higher*, $tables{$day}{$state}->$*->[0]. In other words, because you've got an extra reference in there, you need the extra dereference (${...} or ->$*) before doing the ->[0] dereference to get the array element.

* On Perl 5.20 and 5.22, this feature can be enabled with use feature 'postderef';.

See also perlreftut and perlref.

Update: Minor edits and added second sentence to second paragraph.

#!/usr/bin/perl -CSDA 

use utf8;
use Modern::Perl;
no warnings qw{uninitialized};
use Data::Dumper; 


my $tables = {  # hash ref "{", keys are day numbers
        0 => {  # day0 
                confirmed => [1, 0.334638642561918, 2, 0.8733077785172
+49, 0.8539438, 57158699],
                deaths    => [],
        }, 
        1 => {  # day1 
                confirmed => [2, 0.334538642561918, 2, 0.8733077785172
+49, 0.8539438, 57158699],
                deaths    => [],
        }, 
        2 => {
                confirmed => [3, 0.334638442561918, 2, 0.8733077785172
+49, 0.8539438, 57158699],
                deaths    => [],
        }, 
}; 

$$tables{3}{confirmed} = [3, 0.18, 2, 0.849, 0.8, 59]; 
$$tables{3}{deats} = [2, 3, 4]; 

$$tables{4} = {
        confirmed => [4, 1.3, 2, 1.9, 2, 57],
        deaths    => [],
}; 

$$tables{1}{deats}[3] = "three"; 
$$tables{1}{deats}[5] = "five"; 

$$tables{5}{alfa} = ["a", "l", "faaaaa"];

for my $d (keys %$tables) { 
        print "\nnext day... $d\n";  
        for my $t (keys %{$$tables{$d}}) { 
                print "day is $d, table is \"$t\", content: ", (join "
+, ", @{$$tables{$d}{$t}}), "\n"; 
        } 
} 

print "Dumper is: ", Dumper($tables);
[download]

result:

next day... 4
day is 4, table is "confirmed", content: 4, 1.3, 2, 1.9, 2, 57
day is 4, table is "deaths", content: 

next day... 0
day is 0, table is "confirmed", content: 1, 0.334638642561918, 2, 0.87
+3307778517249, 0.8539438, 57158699
day is 0, table is "deaths", content: 

next day... 2
day is 2, table is "confirmed", content: 3, 0.334638442561918, 2, 0.87
+3307778517249, 0.8539438, 57158699
day is 2, table is "deaths", content: 

next day... 1
day is 1, table is "deats", content: , , , three, , five
day is 1, table is "confirmed", content: 2, 0.334538642561918, 2, 0.87
+3307778517249, 0.8539438, 57158699
day is 1, table is "deaths", content: 

next day... 5
day is 5, table is "alfa", content: a, l, faaaaa

next day... 3
day is 3, table is "deats", content: 2, 3, 4
day is 3, table is "confirmed", content: 3, 0.18, 2, 0.849, 0.8, 59
Dumper is: $VAR1 = {
          '4' => {
                   'confirmed' => [
                                    4,
                                    '1.3',
                                    2,
                                    '1.9',
                                    2,
                                    57
                                  ],
                   'deaths' => []
                 },
          '0' => {
                   'confirmed' => [
                                    1,
                                    '0.334638642561918',
                                    2,
                                    '0.873307778517249',
                                    '0.8539438',
                                    57158699
                                  ],
                   'deaths' => []
                 },
          '2' => {
                   'confirmed' => [
                                    3,
                                    '0.334638442561918',
                                    2,
                                    '0.873307778517249',
                                    '0.8539438',
                                    57158699
                                  ],
                   'deaths' => []
                 },
          '1' => {
                   'deats' => [
                                undef,
                                undef,
                                undef,
                                'three',
                                undef,
                                'five'
                              ],
                   'confirmed' => [
                                    2,
                                    '0.334538642561918',
                                    2,
                                    '0.873307778517249',
                                    '0.8539438',
                                    57158699
                                  ],
                   'deaths' => []
                 },
          '5' => {
                   'alfa' => [
                               'a',
                               'l',
                               'faaaaa'
                             ]
                 },
          '3' => {
                   'deats' => [
                                2,
                                3,
                                4
                              ],
                   'confirmed' => [
                                    3,
                                    '0.18',
                                    2,
                                    '0.849',
                                    '0.8',
                                    59
                                  ]
                 }
        };
[download]

These two snippets are actually equivalent:

my $r = ['a', 'b', 'c'];
[download]

my $r;
{
    my @a = ('a', 'b', 'c');
    $r = \@a;
}
[download]

So when you try using \[1] to get a reference to the anonymous array, you're ending up with a reference to an array reference.

Drop the backslashes n front of the open square brackets: you don't need them. And start to work it out from there.

I am sorry to say that "anonymous array" is actually a misnomer. It's not an array, it's a reference to an array.

Well, if we're going to be nitpicky, then that is not quite correct, and your examples aren't entirely equivalent. Although both $rs are references to arrays, in my @a = ('a', 'b', 'c'); $r = \@a;, it's not an anonymous array, since it has a name, @a (even if its scope is limited). In my $r = ['a', 'b', 'c'];, the array being referenced by $r is anonymous, as it never gets any name. So in the OP's example, it is in fact an anonymous array, albeit a reference to a reference to one.

As soon as the scope is exited, the variable is gone, and the array ref is all that is left. From then on, it's an "anonymized array". The array ref is all that is left.