Having been fixing electronics for a couple of decades now, it's my considered opinion that most things can be repaired.
That being said, information on your particular device seems to be pretty scarce. From what I've been able to determine, the headphone jack is soldered onto the motherboard, so soldering skills will be required in order to replace it. I didn't have any luck finding any replacement parts outside of batteries, and certainly no guides as to how to replace your jack.
I did find a teardown video that basically consisted of showing you how to remove the motherboard, which is good, because that's exactly what you'll have to do to fix it. Here's the YouTube video.
How to Disassemble the FiiO X1 - YouTube
If that's something you want to attempt, we'll be happy to work with you to figure out what part you need and how to do the replacement. I'd say the first step would be to follow that video and get the motherboard out. Note that there's a couple of steps that I'd suggest doing differently; the author of the video removes several screws as the first step, but I'd make the first step that of unplugging the battery. Also, they start unplugging and disassembling only to find out that they've missed a screw and have to go back and remove it before they can proceed, so watch the video through and make sure you get all the screws before moving on.
From some of my reading on forums, it appears you need a Pentalobe screwdriver; that's one with five points. iFixit sells them in different sizes, but I have no idea which one you'll need for your DAP.
Once you have the motherboard out, take pictures of the front and back sides of the headphone jack; from there we can try to identify a replacement that will work. The higher the resolution and the better the light, the easier it will be for us to identify a replacement part.
If this kind of repair is beyond your comfort zone then you'll want to check with local electronics repair shops and see if they're willing to give it a try. It should be a relatively simple repair for a professional technician; the only sticking point might be in finding and ordering the jack, but a decent shop should be able to figure it out.
Let us know what you think; if you want to try it on your own, that's what this whole site is about and we'll be glad to work with you to get it fixed.