Compatibility.

The new Apple Watch Series 3 and 5 shipping from Apple come with watchOS 6 and require an iPhone 6s, SE, 7, 8, X, XS, XR, 11 or 11 Pro, running iOS 13

The 5s is not supported.

This is specified on the Apple webpage for the Apple Watch in the footnotes at the bottom of the page.

A brand new, never opened Apple Watch Series 4 will have watchOS 5 on it, and should work with the iPhone 5s (but I think that is a GPS only version; I think the Series 4 cellular needs an iPhone 6 or newer, but I am not really sure about that).

Since Apple has stopped manufacturing the Series 4, you will have to get it form a retail stores existing inventory.

If you are talking the used Apple Watch market, then you have to A) be careful the previous owner did not upgrade to watchOS 6, B) that the previous owner has followed these instructions:

What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else

https://support.apple.com/en-us/HT208855

if they have not followed those instructions, then make sure the previous owner can be contacted, otherwise you have a useless Apple Watch. NOTE: Many online Craig's List, eBay kind of used sales are either done by people that do not follow the instructions, or are stolen Apple Watches and the seller will Ghost you as soon as they get the money.