Want to replace 2013 27 inch iMac with solid state computer

Well, Apple seems to have made the 27-inch iMac a "non-thing," so you can look at Apple refurbs (from Apple ONLY) or a Mac Studio or Mac Mini with a free-standing 27-inch monitor.

https://www.apple.com/shop/refurbished/mac/27-inch-imac

Some better monitors have cameras and speakers built in so would duplicate the functions of your current 2013 model.

Welcome to the Apple Support Communities.

In addition to the other posters, you could get a Mac mini - Apple or a Mac Studio - Apple, with a Studio Display - Apple, or look here for 27-inch iMac computers: Certified Refurbished Products - Apple.

Another option is to setup an external SSD on your current iMac. You'll run macOS and your data from it, bypassing the failing drive: How to Setup and Use an External SSD as y… - Apple Community.

Regardless, since your disk is failing, make sure you have a backup. If you don't, you should create one. To start, you need an external hard drive. Here’s a few excellent ones sold by Apple: 

• LaCie Mobile Drive 4TB External Hard Drive USB-C - Apple
• LaCie 4TB Rugged USB-C Portable Hard Drive - Apple

If you have an iMac (Late 2015) or earlier, or a Mac mini (Late 2014) or earlier, you will also need this cable. Since you have an iMac (21.5-inch, Late 2013), you will need the cable for one of the above drives. 

You can then follow the steps in this Apple Support article to setup your Time Machine backup: Back up your Mac with Time Machine - Apple Support. 

Also, moving forward, I recommend that you use iCloud in addition to Time Machine to sync all your files and photos to all your Apple devices. Although iCloud is not a "true" backup service, iCloud can also act as an emergency off-site backup, but this is why iCloud should be used in addition to Time Machine. Here are some resources on setting up iCloud for Mac:

• Upgrade to iCloud+ - Apple Support
• Set up iCloud Drive - Apple Support
• Add your Desktop and Documents files to iCloud Drive - Apple Support
• Set up and use iCloud Photos - Apple Support

After you setup iCloud, turn off "Optimize Mac Storage" in System Preferences > Apple ID/iCloud. That way, all changes made to files in iCloud are also backed up to Time Machine. If you have this setting on due to a limited amount of storage on your Mac, that is fine. Just know changes won't be backed up to Time Machine, only iCloud.

Jack

I'm looking towards a Mac Mini with a Studio display, but I'll wait till the Mini gets an update. My 2013 27" iMac is still running good.

There's still a big gap price wise between the new iMacs and even a low end Mini/Studio Display setup. I'm still hoping they'll fill the gap with a larger iMac eventually. By the look of the posts here there are quite a few more people looking for a bigger iMac.

What is your question?