Although either is possible, both seem unlikely. In my personal experience (decades doing repair work) a PS rarely fails without giving some kind of advance indication. To rule it out, see if you can borrow one from a friend. Most PS are fairly standardized, and unless you go for something elaborate, not too expensive.
Before you get into changing parts, however, there is another culprit I have seen many times, especially in systems rarely used: loose add-on cards. If your computer has a separate video card (instead of on-board) take it out and "re-seat" it back in again. Then try the boot. If that makes no difference, do the same with any other add-on cards.
It's also useful to note the EXACT point at which the system reboots. Is it always exactly the same point in the start up cycle, or does it vary? That can help find the issue if you can determine what is happening at the point of failure. Be sure to try and get into Safe Mode if you can.
Assuming all the above fails, there is a free Windows utility called Hiren's Boot CD which is one of the most useful I've found for diagnostic work. It boots your computer through one of the usual Linux-loaded Windows XP implementations. The point of this is since the Windows OS is not loaded, you'll at least find out if its the PC that's going down, or the OS (Windows).
You'll need to set your boot priority in your BIOS for "CD First" and of course, you'll need a friend's PC to download and make the startup disc for you. Just Google "Hiren's Boot CD" and you'll find it.