Depends on what you'd like to store and how. In theory with Git you should be able to define a .gitignore file that is appropriate and then set your entire JDE directory on a development client as a git repository. Then you would only need to create a PAR and store that for major changes to the relevant objects but you could store incremental code changes as source. In terms of which service to use, that is up to you. You can either create your own private Git server on the network or use a cloud solution. If you're looking at cloud then I'd recommend bitbucket because if you only have small development team (less than 5) you can have unlimited private repositories at no cost.
The one thing to be cautious about is the "sharing of JD Edwards objects" - obviously sharing of Oracle proprietary code written in the Event Rule Language is a BIG no-no - the issue, of course, is that so much code references "vanilla" JDE objects, that you're in danger of sharing something that is specifically covered by your legal agreements. If you share an object to a public cloud, then please be sure to ensure that NO vanilla code is shared. Many times, "custom" objects are created by a user copying existing vanilla objects and then modifying them - if you distribute a substantial amount of the original object, you'll likely get a phone call from Oracles' attorneys - and nobody wants that !
Also, if you create code for a customer, and then share that code - you might also be in breach of intellectual property rights, even if the code seems completely minor.
There have been a number of times in the past 15 years where people have thought about having a "central" library of custom routines and code for customers to easily implement. I've even asked Oracle a couple of times to come up with their own "app store" to no avail. There are so many times that developers have to create similar reports between multiple customers - and it seems useful on the face of it. However, the reality will be that as soon as something gets published like this, Oracle will look at every line of code and the risk is substantial to the customers and the developers. Remember SAP and TomorrowNow ?! (admittedly that is an extreme example with a competitor - but it is still a baseline for everyone to consider !)