Earier Imac doesn't complete startup. Stuck on logo and progress bar

Please refer to If your Mac doesn't start up all the way - Apple Support. It is likely a hardware failure. What model year is the old iMac?

A 2013 is considered obsolete and if it has any type of HD in it that has likely failed. IMHO it is not worth fixing, however you can take it to your local independent AASP to have it diagnosed.

Here is the process to locate Apple Authorized Service Providers:

1. Go to Apple Support.
2. Follow the prompts to make an appointment for your phone screen repair.
3. Then, you will see a list of Apple Authorized Service Providers near your area. Additionally, you will see other repair service providers, including third-party repair services.

Another option is to consider replacing it. You have a LOT of options available to you and if you want some help to explore those feel free to ask.

Shut down your Mac, then using a wired keyboard, startup into Internet Recovery Mode (Option-Command-R) by immediately pressing and holding those keys at startup. Then, please go to Disk Utility, and select "View" > "Show All Devices" (if shown). Select each device (or entry shown), and take a photo of it and upload them all here. 

Jack

Follow the steps at > https://support.apple.com/en-us/HT204156 for the failed startup screen that you see.

Your welcome and good luck.