If I recall correctly, I have seen people report that they could install and run Office 2003 in Windows 8. So, it likely will not hurt to try installing it and seeing how it runs. It might run well it enough, but you might encounter some issues. Microsoft officially lists it as not being compatible.
Here is a Microsoft Answer thread that has a few people saying they got it to work…note this was with Windows 8, not Windows 8.1:http://answers.microsoft.com/en-us/office/forum/office_2003-office_install/will-office-2003-work-with-windows-8/b836c7d6-7b01-4f46-8764-d1f00c53186d
Personally, I would likely go with a newer version. Which one to go with is somewhat more of a complex issue.
First, how many computers would need/want to run it on now? How likely would that be to change in the future?
Are you the type of person who does not like/want to upgrade often (that seems to be be the case since your old computer was still running XP and Office 2003)? Or do you like/want/need to have the latest and greatest?
What specific features do you want/need out of Office? Which Office applications do you need (i.e. do you need Access or Publisher or just Word, Excel, and Powerpoint)? And is this for business use (i.e. using it to make money) or home use (not using it to make money)?
Generally, if you only need to run it on one computer and don't have a need/want to be using the latests and greatest, then I would avoid Office 365. Office 365 becomes a good deal when you need to run Office on multiple computers and want to keep using the latest and greatest versions of Office. Otherwise, a single Office 2013 (or Office 2010…if you can find it) license will likely be cheaper in the long run. You can get a single "business use" (i.e. for making money) license for Office 2013 for a little over $200:http://www.amazon.com/Microsoft-Office-Business-2013-Download/dp/B00B1TGMIS/ref=sr_1_4?ie=UTF8&qid=1390626558&sr=8-4&keywords=office+2010
Or you can get a single "personal use" (i.e. NOT for making money) license for Office 2010 for a little under $100:http://www.amazon.com/Microsoft-Office-Home-Student-2010/dp/B0039L31JY/ref=sr_1_1?ie=UTF8&qid=1390626640&sr=8-1&keywords=office+2010
You should note that Office 365 does give you some benefits beyond just use of the Office suite such as 20 GB of Skydrive space, 60 Skype minutes, and you get access to ALL Office apps, which stand alone version may only give you the basics such as Word, Excel, and Powerpoint (depends on the version you get). But, if you are not the type to upgrade to new version frequently, don't need it on more than one device, and don't need all the Office apps (just the basics), then it costs more over the long haul…and since it is subscription based, the moment you stop paying, you lose the ability to run the local, installed Office apps.
Your other option would be to consider one of the free office suites such as Open Office or LibreOffice.