recovery server cannot be contacted

Do you have or have access to another Mac (laptop perhaps etc) that you can use to download the installer? If yes then you can use TDM (target disk mode) with an appropriate cable to reinstall the OS on your iMac. If you manage to do this I would invest in an externally attached SSD drive that is a clone if your iMac which you can use in circumstances like the one you describe.

Target Disk Mode instructions here:

Transfer files between two Mac computers using target disk mode – Apple Support (UK)

Download Sierra OS here:

How to download macOS - Apple Support

Click on the link for Sierra and install it. Once installed use the high Sierra link to the App Store to upgrade to high Sierra.

If you manage to do this successfully then reinstall any additional apps and devices you may need.

Good luck

First thing to try is to get closer to your Wi-Fi source as you're probably too far away. IE: be in the same room as your router. Better still use a wired connection instead as this is always more reliable. There should be an ethernet cable that came with your router that you can use. Depending on your model computer you may need an adapter.

Potentially the problem may lie with the quality of your internet service. Fine for web browsing in the main but not quite robust enough for installing the OS. Try downloading the dog instead.