Oracle does like to sell their software - and it sounds like they have an upgrade methodology that supports Goldengate. If so, then that IS one way to upgrade - but its not going to be cheap.
Theres quite a bit of information you left out in your request (ie, what tools release you're on, etc etc). And the following is a little open to lots of interpretation by peers about whether its "safe" or not. The thing is, you're on "ancient" software already - and you're trying to jump a bunch of versions with this upgrade - which is exactly why its always recommended to be no more than 5 years behind versions on your ERP system ! (sorry for the little dig there - but you DID save a ton of cash by NOT going through upgrades over the past 10 years !)
SO, your alternative would be to upgrade the OS and SQL Server to the most recently supported versions for your 8.10 install and THEN upgrade it to 9.1 and THEN perform a second OS and SQL Server upgrade to get current.
I was looking at the MTR document (Minimum Technical Requirements have been superseded with "certifications" for newer versions) - but there are historical documents out there that will help.
I checked some of the older documentation - and TECHNICALLY the last supported Tools Release for 8.10 is 8.98.1
According to Document ID 705411.1 - support for SQL 2008 x64 (not R2) began with JD Edwards EnterpriseOne 8.98.1 (Note that you will also have to implement a planner ESU to achieve this.)
According to the certification, 9.1.2 supports SQL 2008 x64 as well - so you have a path !
Theoretically, a tools release is not a big impact on an E1 system - but in reality it can dramatically change certain aspects of your functional code and integrations - so you WILL need to VERY THOROUGHLY test.
IF you manage to get 8.10 with 8.98.1 up and running, then upgrading the OS and database should not be too difficult (but might be time consuming - again, backup and test !)
Now, once you're running 8.10 on 8.98.1 tools against Win/SQL 2008 x64 - you're in a very good position, since 9.1 will support Microsoft SQL Server 2008 R2 and Windows 2008. You'll do your 8.10 to 9.1 upgrade on SQL 2008 and then some time later you can later upgrade your database to SQL 2014 and whatever OS version you want too.
You could try this on a simple sandbox environment (grab three workstations with some decent storage and make them a deployment server, enterprise server and development workstation). Install your 8.10 code onto it with your current TR, and then attempt the 8.98.1 Tools installation (should take a couple of days or so). If that works, then you've got over the "difficult" bit. You can then perform a Windows 2008/SQL 2008 upgrade (relatively simple), then perform the 9.1.2 upgrade (difficult), then when you've ensured everything is running, perform a Windows 2014/SQL 2014 upgrade. The entire process shouldn't take more than a week or two to test out on a sandbox.
NOW - my disclaimer is that the above doesn't help you with any custom code, external interfaces and all the other gubbins that the average implementation has. You might be able to upgrade E1 in that manner, but you might have issues with other software that E1 talks to.
Good luck to you ! I'll be very interested to hear if you manage to pull it off.