Thanks! I got it to work on CentOS7. Here are my steps:

sudo su -
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Create /etc/yum.repos.d/unit.repo:

[unit]
name=unit repo
baseurl=https://packages.nginx.org/unit/centos/$releasever/$basearch/
gpgcheck=0
enabled=1
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Install unit and unit-perl

yum install -y unit unit-perl
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In /etc/unit/ create two files:

start.json

{
   "type": "perl",
   "processes": 5,
   "script": "/etc/unit/app.psgi"
}
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app.psgi

use strict;
use warnings;
use Plack::Builder;

my $app = sub {
    my $env = shift;
    [200, [], ["Hello world from PSGI!\n"]];
};

builder {
    enable 'ContentLength';
    $app;
};
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Then start unit:

service unit start
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Configure unit using the following commands:

curl -X PUT --data-binary @/etc/unit/start.json --unix-socket /var/run
+/control.unit.sock http://localhost/config/applications/psgiapp
curl -X PUT --data-binary '{"application":"psgiapp"}' --unix-socket /v
+ar/run/control.unit.sock 'http://localhost/config/listeners/*:8000'
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.. and check the resulting config:

curl --unix-socket /var/run/control.unit.sock http://localhost/
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... and finally do the Hello World test:

curl -v http://127.0.0.1:8000/
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Voila!