My iMac will not boot up. It failed to download the new High Sierra and crashed

Startup in macOS Recovery > select Disk Utility > Repair the Macintosh HD startup disk > then go to Reinstall MacOS and install MacOS.

see > About macOS Recovery - Apple Support for the full instructions.

If your iMac will not startup in macOS Recovery or if it is an older iMac from the original Install disc, then you need to seek help for your local Apple Store or local Apple Authorized Service Provider.

https://locate.apple.com/

It keeps telling me that there is not enough memory to install the OS. Retry and the cycle starts all over. This last time, I got an exclamation point over the globe and underneath it says "apple.com/support" and under that it says "-1005F"

Me too. Looks like others are having the same issue. So my mac mini is now in the shop where it will cost me $120 to try to get it to work. Otherwise it's a restore from my backup. It fails to boot into anything. Spent an hour on the phone with Apple about it and there was no fix. Not pleased at all with Apple. My mini is only two years old. I've sent out the word to others I know who have Apples NOT to do the download without a full backup beforehand and to be prepared for trouble.

This answer is only helpful if you can get your computer into Recovery mode. Mine will not open in Disk Utility nor will it even try anymore to reinstall the OS. It's in the shop for a costly evaluation and fix.

I was able to boot my macbook in target disk mode, create more room (at least 15gb) by deleting the files created by the prior attempt to install High Sierra, and then it would reboot in recovery mode. Then I chose to install a new copy of High Sierra from the recovery menu, and all is well again. I'm sorry you had to take yours to Apple, I feel fortunate that I was able to do it myself.

My computer will not reboot so I can’t do that? Not helpful.

This is very painful. I have downloaded this OS 3 times and yet it still fails. I will be very very cautious in installing updates going forward. Poor change management from Apple 😠.