The raspberry pi is connected through ethernet to the router and I can check/download updates without any issues.
This is the screen I always get when attempting to install Octoprint:

What happens if you try the above (outside of dietpi-software)?root@VM-Stretch:~# G_CHECK_URL https://github.com/foosel/OctoPrint.git
[ OK ] Connection test: https://github.com/foosel/OctoPrint.git
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root@livingroom:~# cat /etc/resolv.conf
# Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8)
# DO NOT EDIT THIS FILE BY HAND -- YOUR CHANGES WILL BE OVERWRITTEN
nameserver 192.168.1.1
root@livingroom:~#
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root@livingroom:~# curl https://github.com/foosel/OctoPrint.git
<html>
<head><title>301 Moved Permanently</title></head>
<body bgcolor="white">
<center><h1>301 Moved Permanently</h1></center>
<hr><center>nginx</center>
</body>
</html>
root@livingroom:~#
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su dietpi #octoprint needs to be later started by non-root user
cd
sudo apt-get install git python-pip virtualenv
git clone https://github.com/foosel/OctoPrint.git
cd OctoPrint/
virtualenv venv
./venv/bin/python setup.py install
./venv/bin/octoprint #starting of octoprint