It sounds like you’ve got a mature perspective, but you’re only 22, so you’ve got plenty of time. You’re sufficiently dissatisfied with your job that you want to make a change, but it doesn’t sound like you hate it so much that it’s killing you (e.g. you’re super-stressed and get physically sick because of work). Further assuming that you don’t have a mortgage (or other significant financial ties) and you don’t have dependents, you’re in an excellent place to make a major change in your life.
I’m going to largely sidestep the whole “can I make a living in the arts?” issue because it’s such a huge and contentious topic. But I would certainly encourage you to not just repress your ambitions simply because they’re “unlikely to work out” - whatever that means. There is not much in life I’m certain about, but I am pretty certain that ignoring your ambitions on that basis alone is a recipe for disappointment later in life.
solipse’s comment about investigating cheaper options into your desired field is certainly one worth investigating. But so is the HE route. As Kaiju hinted at, going to uni gives you time to develop yourself in a supportive environment (at least that’s what I’m hoping, since that’s my plan for when I start at uni – hopefully – in 2015). You may find that if you have to stay too long in your Civil Service job, your passion, while never disappearing completely, will get eroded (I’ve been in exactly that position myself in the past). So, the “pursue your music/graphic design on the side while holding down the day job” approach is certainly valid, but it may not be the best option for you.
Have a look around at HE courses and see what grabs you. If you do decide that it is the right route for you, as a general principle, my advice would be to choose a course that really interests you, rather than trying to endlessly second-guess “what the marketplace wants” and base your choice on that. Giving up your soul-destroying job to study a subject you’re interested in is a positive move; giving up a comfortable salary to do an expensive course that you don’t enjoy and which keeps you too busy to earn a decent income is, I’d suggest, a negative move. (I’m not an expert on breaking into the music or graphic design professions, so I freely admit I may be wrong on this point.)
A further comment I’d add, if you do decide HE is the way forward for you, is that you may want to act a little more cautiously than coffeecakey has urged you to. HE is an expensive option. If you do find out you also need to do an Access or some other preparatory course to get into uni, finances may become even more of an issue. For Access courses at least, while you can get a loan to cover the cost of the course itself, you can’t get much (if anything) in the way of institutional help with living expenses. And Access courses are, if you choose the one year route, a full-time commitment – you wouldn’t be able to do it while keeping your current f/t job. So the challenge you have to overcome is how to keep up with the demands of the course while still being able to eat. Based on the assumptions I made in the first paragraph of my post, you have a job that is tolerable for you in the short or medium term as long as you’re simultaneously plotting your escape route. I’d suggest getting started on that escape plan right away, but carefully consider the form it will take. Meanwhile, use your earnings to build up as much of an HE savings fund that you can.
I’d say it’s better to tread cautiously but confidently than to dive into this half-arsed. But if you do decide that you want to go to uni in 2015 and that’s manageable…fine, go for it. However, what I would urge you to do is take positive action. Don’t fret about this so much that you don’t do anything at all. I can virtually guarantee there will be naysayers at your current workplace who will cast doubt over your ambitions, but you have to be strong. Keep moving towards your goal and don’t let them put you off. Good luck!